The Inlaid Jewels Tantra, for example, rejects the Yogācāra definition, stating:  

"Untainted vidyā is the kāya of jñāna (tib. ye shes). Since svasaṃvedana (rang gyis rig pa or 'rang rig') is devoid of actual signs of awakening, it is not at all the jñāna of vidyā (rig pa'i ye shes)."

Ju Mipham states regarding the substantialist Yogācāra view in Liquid Gold:

"The Cittamatrin Yogācārins deconstruct both subject and object in a mere empty intrinsically knowing gnosis (jñāna)."

The difference between that svasaṃvedana of Yogācāra and the svayaṃbhūjñāna of ati is, as he says:  

"When the pairing of the dhātu and vidyā is deconstructed, there is no focal point upon which to grasp. Once it is understood that the final premise, “this is ultimate,” is deconstructed in the state of inexpressible emptiness, one enters into the nondual jñāna (tib. ye shes) that all phenomena of the inseparable two truths are of the same taste."

Longchenpa writes regarding the Yogācāra view that Dzogchen even rejects that dharmatā is "nondual," he says: 

།གང་ལ་གཟུང་བ་དང་འཛིན་པ་མེད་པར་རྟོགས་པའི་རིག་པ་དེའི་ངོ་བོ་ལ་ནི་རང་བྱུང་གི་ཡེ་ཤེས་སུ་ཐ་སྙད་བཏགས་ཀྱང༌། རང་རིག་རང་གསལ་ལོ་ཞེས་རྣལ་འབྱོར་སེམས་ཙམ་པ་ལྟར་མི་འདོད་དེ། ཕྱི་ནང་མེད་པས་ནང་གི་སེམས་སུ་མ་གྲུབ་པ་དང༌། རང་གཞན་མེད་པས་རང་གི་རིག་པ་ཁོ་ནར་མ་གྲུབ་པ་དང༌། གཟུང་འཛིན་ཡོད་མ་མྱོང་བས་དེ་གཉིས་དང་བྲལ་བར་མ་གྲུབ་པ་དང༌། ཚོར་རིག་གི་ཡུལ་ན་མེད་པས་མྱོང་བ་གཉིས་མེད་དུ་མ་གྲུབ་པ་དང༌། སེམས་དང་སེམས་བྱུང་མེད་པས་རང་གི་སེམས་སུ་མ་གྲུབ་པ་དང༌། གསལ་མི་གསལ་དུ་མེད་པས་རང་གསལ་དུ་མ་གྲུབ་པའི་ཕྱིར་རོ། །རིག་མ་རིག་ལས་འདས་པས་རིག་པ་ཙམ་དུའང་གདགས་སུ་མེད་པ་འདི་ནི། མཐའ་བྲལ་ཡོངས་སུ་རྫོགས་པ་ཆེན་པོ་ཞེས་བྱ་སྟེ། མཚོན་ཚིག་གི་ཐ་སྙད་རང་བྱུང་གི་ཡེ་ཤེས་དང༌། བྱང་ཆུབ་ཀྱི་སེམས་དང༌། ཆོས་སྐུ་དང༌། དབྱིངས་ལྷུན་གྲུབ་ཆེན་པོ་དང༌། རིག་པ་རང་གསལ་རྗེན་པ་ཞེས་བརྗོད་ཀྱང༌། བརྡ་ཤེས་པའི་ཕྱིར་བཏགས་པ་ཙམ་ལས་རང་ངོ་བརྗོད་མེད་ཆེན་པོར་རྟོགས་པར་བྱའོ། །དེ་ལྟར་མ་ཡིན་པར་མིང་ལ་དོན་དུ་ཞེན་ནས་སེམས་ཙམ་པའི་རང་རིག་རང་གསལ་གཟུང་འཛིན་གཉིས་མེད་ཀྱི་ཤེས་པ་དང་ཁྱད་པར་མི་རྙེད་དོ།  

"Though the essence of knowledge (rig pa) that realizes there is nothing apprehended or apprehending is conventionally designated “self-originated pristine consciousness,” rang rig rang gsal is not asserted in the way of the Cittamatrin Yogacārins [svasaṃvedana] because (1) since there is no inside or outside, the inner mind is not established; (2) since there is neither self nor other, a reflexive knowing (skt. svasaṃvedana, tib. rang gyi rig pa) is not established at all; (3) since there is no apprehended object or apprehending subject, freedom from duality is not established; (4) since there is no object to experience, experience is not established as nondual; since there are no minds and mental factors, one’s mind is not established; (5) since there is neither clarity (gsal ba) nor absence of clarity (mi gsal ba), intrinsic clarity (rang gsal) is not established. (6) Because of being beyond knowing or unknowing, even knowing does not exist as a designation—this is called “the great total perfection beyond extremes (mtha’ bral yongs su rdzogs pa chen pa).” Though illustrative conventions are expressed such as “self-originated pristine consciousness,” “bodhicitta,” “dharmakāya,” “the great naturally perfected dhātu,” and “naked, intrinsically clear cognizance (rig pa rang gsal),” other than being mere terms for understanding symbols, the real nature must be realized as a great inexpressibility.  

Otherwise, there is no difference at all with the Cittamatrin’s self-knowing and self-illuminating consciousness devoid of an apprehending subject and an apprehending object through clinging to meaning in a name."

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Lopön Tenzin Namdak explains the samaya of the basis called gcig pu which represents a generic characteristic (samanyalakṣaṇa):

"That is Chigpu (gcig pu) - without any partition. It means that each individual being has a mind and the nature is of a very similar quality.

Don't think that there is just one nature (for everyone). Don't think it is like the sun, that there is just one sun but its rays cover everywhere. Each being has mind and wherever there is mind, there is nature - it is not separate from mind but nature is not just the same (one). Each individual being has nature and this nature is practiced and realized by the individual; it is the individual who takes the result.

When the text says Thigle Nyagchig, it means similar quality; emptiness, clarity and unification are the same everywhere.

For example, if you cut down one stick of bamboo you can see it is hollow and so you don't need to cut down all the bamboo. In a similar way, if you realize (the nature of your mind) it is your mind which liberates in to nature. All sentient beings who have mind are integrated with nature. That is Thigle Nyagchig. That is what single means.

If you depend on consciousness, that is breaking the Dzogchen vow (damstig). That is the main thing."

[...]

"If you don't understand this clearly but think that one mind pervades everything, then that is what is kept and learned in Vedanta; that is their very strong view. If you believe this then your damstig is broken and you go against the meaning of Dzogchen. Is that clear? You must make sure (of this point). If you think that (nature) is one with individual partitions, that this "one" pervades everything, then that is breaking your Dzogchen damstig and goes against the Dzogchen view. Hopefully you have understood clearly."

In a conventional sense, each conventional Buddha has their own mind (citta), and each mind has its own nature (citta dharmatā) that is intended to be recognized. Dharmakāya is a buddha's citta dharmatā, or *cittatā.* Which means dharmakāya is the dharmatā of a buddha’s mind. The dharmakāya is a buddha's jñāna.  

We can say that each conventional buddha has their own conventional gnosis (jñāna), because ultimately there is no jñāna, and no dharmakāya. The characteristic of jñāna is a buddha's knowledge of emptiness. The dharmakāya is a buddha's realization of emptiness, which is known individually.  

In the yogic direct perception (yogapratyakṣa) of emptiness, there are no distinctions between phenomenal entities because entities cannot be found, hence like the Śrīmāladevi says, the dharmakāya is the "space-like gnosis of the tathāgatas."  

The generic basis (spyi gzhi) is just a generic set of qualities, essence (ngo bo) and nature (rang bzhin), which are characteristics that all minds possess.

The foundational structure for understanding this topic is somewhat elaborate, and can't be communicated in a brief post - or even a single post since there are many factors that need to be taken into consideration. However, the "individual and universal" are found to be complimentary when understood to be descriptions of what is called a "generic characteristic" (samanyalakṣaṇa). The dharmakāya is the nature (dharmatā) of a mind. That nature, or dharmatā is a generic characteristic, which is an abstraction.  

You'll often see statements such as the dharmakāya is "neither one nor many," and this is easily misunderstood. The intended meaning however is that as a generic characteristic, the dharmakāya is not "one," because it is present in countless individual minds wherever those minds are found, and it is not "many," because wherever it is found, it is identical in expression. Similar to the heat of fire. Heat is also "neither one nor many," it is not "one," because it is present in countless individual instances of fire wherever fire is found, and it is not "many," because wherever heat is found, it is identical in expression. The dharmakāya is the same, for example, the *Mahāyānasaṃgraha* says:  

"Likewise, the dharmakāya has the characteristic of the nonduality of oneness and difference [it is not one nor many] because the tathāgatagarbha is not different [in expression], while innumerable mind streams reach fully perfect awakening [individually]."

The dharmakāya is the generic characteristic of an individual mind. In Dzogchen teachings for example, the jñānas of ka dag and lhun grub are called the "generic basis" or spyi gzhi because they are the essence (ngo bo) and nature (rang bzhin) of thugs rje, which is the instantiation of an individual consciousness. This means that ka dag and lhun grub are generic characteristics, and the basis is therefore not a real essence. Not actually established in any way. 

In Dzogchen teachings we don't really speak of conditioned and unconditioned phenomena, but the principle still applies. What is essentially being said, is that the dharmakāya, which is unconditioned, is the generic characteristic (samanyalakṣaṇa) of the conditioned, one's mind.  

The point being made here is that one's mind has always been so-called “unconditioned” dharmatā, the dharmakāya, from the very beginning, however due to delusion, this is not recognized, and we mistakenly conceive of an individual mind even though there is no such thing.  

The mind is therefore unreal from the very beginning, but due to our confusion, we mistakenly conceive of a mind, and as a result, we must endeavor to recognize that the mind is actually insubstantial and not established in any way. We call that essenceless nature, the mind's dharmatā, the dharmakāya, but since there is no actual mind to have a nature, there also is no actual nature. The dharmatā of the mind is simply something to recognize about the mind, and once we recognize this, then it is seen that there never was a mind in the first place to even possess a nature.  

Buddhas and realized beings do not see the allegedly conditioned mind as a conditioned "mind," because they know the true nature of what we mistakenly conceive of as "mind." As the Lokadharaparipṛcchā says:  

"Lokadhara, it is not the case that unconditioned phenomena exist separate from conditioned phenomena, or that conditioned phenomena exist separate from unconditioned phenomena, for the characteristic of the suchness of the conditioned is the unconditioned. Why is this? There is nothing conditioned within the conditioned, and nothing unconditioned within the unconditioned."

This theme is also found in Dzogchen wherein there is actually nothing conditioned in the allegedly conditioned, and since the conditioned cannot be established, the unconditioned cannot be established, like Nāgārjuna says.  

The Six Dimensions says:  

"Dharmatā free from proliferation is originally pure; it is the basis of an intrinsically pure nature; it is free from words and syllables; it cannot be confirmed through expression; it is free from all conventional reification; it is without concepts of apprehended objects and apprehending subjects; it is without buddhas and without sentient beings; it is without phenomena and without perception of phenomena; no one, no thing, nothing at all. When the essence of such nonexistence (med pa) is confirmed with some words: the essence (ngo bo) is original purity (ka dag) and the nature (rang bzhin) is natural perfection (lhun grub)."

The Rig pa rang shar rejects Advaita Vedanta, mentioning Ādi Śaṅkara by name.

Vimalamitra states:

The basis, the state of initial original purity, is liberated because its essence is not established at all.

The Mind Mirror of Samantabhadra:

Since there is no ultimate, also the name “relative” does not exist.

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And then all of these teachers stating that Dzogchen is compatible with the Madhyamaka view and emphasizes emptiness, which obviously undermines something like Advaita Vedanta.

From Khenpo Tsultrim Gyamtso:  

"Furthermore, since one must rely on Nagarjuna’s reasonings in order to realize the essence of Dzogchen, it is the same for Mahamudra. Those who studied at the shedras (philosophical universities) in Tibet studied *The Fundamental Wisdom of the Middle Way* and Chandrakirti’s *Entering the Middle Way* and other similar texts over the course of many years. Mahamudra and Dzogchen were not studied, however, because it is the Middle Way texts that are filled with such a vast array of different arguments and logical reasonings that one can pursue the study of them in a manner that is both subtle and profound. In the Mahamudra teachings as well, we find statements such as this one from Karmapa Rangjung Dorje’s Mahamudra Aspiration Prayer:  

'As for mind, there is no mind! Mind is empty of essence.' 

If you gain certainty in mind’s emptiness of essence by analyzing it with the reasoning that refutes arising from the four extremes and with others as well, then your understanding of Mahamudra will become profound. Otherwise, you could recite this line, but in your mind it would be nothing more than an opinion or a guess.

If you study these reasonings presented in 'The Fundamental Wisdom of the Middle Way', when you receive Mahamudra and Dzogchen explanations of emptiness and lack of inherent reality, you will already be familiar with what is being taught and so you will not need to learn anything new. Mipham Rinpoche composed a brief text called 'The Beacon of Certainty', in which he states: 'In order to have perfect certainty in "kadag" (primordial purity) one must have perfect understanding of the view of the Consequence or Prasangika school. Kadag, or original, primordial purity, is the view of Dzogchen, and in order to perfect that view, one must perfect one’s understanding of the Middle Way Consequence or Prasangika school’s view. What this implies is that the view of Dzogchen kadag and the view of the Consequence or Prasangika of Chandrakirti's school are the same."

From Tulku Tsullo's instruction on the view of Dzogchen:  

"Therefore, whether in sutra or in tantra, there is consensus that the only direct antidote to the ignorance of clinging to things as real - which lies at the root of our karma and disturbing emotions - is the wisdom that realizes emptiness. So for Dzogchen practitioners, too, it is extremely important to realize emptiness."

The sgra thal gyur tantra states:  

"Nonexistent therefore appearing, appearing therefore empty. The inseparable union of appearance and emptiness with its branches."

Zilnon Zhepa Tsal said:  

"How could liberation be attained without realizing emptiness? And how could emptiness be realized without the Great Perfection [Dzogchen]? Who but I offers praise such as this?"

The Dalai Lama states:  

"We need a special form of wisdom - the wisdom that realizes emptiness - to act as the direct antidote to the cognitive obscurations. Without this wisdom, which can be realized through the Great Perfection... we will not have the direct antidote to the cognitive obscurations. So this point is conclusive."

Khenchen Rigdzin Dorje [Chatral Rinpoche's heart disciple] states:  

"The Madhyamika consider the Prasangik as the perfect Rangtong view. The Dzogchen trekcho view as Kadag (primordially pure view) and the Prasangik view is the same. The emptiness is the same, there is no difference... It is important to understand that the words primordially pure [kadag] is the Dzogchen terminology for the Prasangic Emptiness. [The ancient Nyingmapa Masters like Long Chenpa, Jigme Lingpa, Mipham, were] Prasangikas [Thalgyurpas]... the Prasangika Madhyamika sunyata [tongpanyid] and the Dzogchen sunyata are exactly the same. There is no difference. One hundred percent [the] same."

Longchenpa says:  

"This system of the natural great perfection is equivalent with the Consequentialist [Prasangika] Madhyamaka’s usual way of considering freedom from extremes and so on. However, emptiness in Madhymaka is an emptiness counted as similar to space, made into the basis; here [in Dzogchen] naked pellucid vidyā pure from the beginning that is not established; that, merely unceasing, is made into the basis. - The phenomena that arise from the basis are apprehended as being free from extremes, like space."

David Germano:  

"While a detailed analysis of the relationship of these classical Great Perfection texts to the Madhyamaka Prasangika tradition is quite beyond the scope of my present discussion, at this point I would merely like to indicate that even in The Seventeen Tantras (i.e. without considering Longchenpa's corpus) it is very clear that the tradition embodies an innovative dialectical reinterpretation of the Prasangika notions of emptiness, rather than a mere sterile 'diametric

opposition' to them that Karmay suggests."

Ju Mipham Rinpoche states in his commentary on the Madhyamakālaṃkāra, the dbu ma rgyan gyi rnam bshad 'jam dbyangs bla ma dgyes pa'i zhal lung:  

"Without finding certainty in primordial purity (ka dag), just mulling over some 'ground that is neither existent nor nonexistent' will get you nowhere. If you apprehend this basis of emptiness that is empty of both existence and nonexistence as something that is established by its essence separately [from everything else], no matter how you label it (such as an inconceivable self, Brahmā, Viṣṇu, Īśvara, or wisdom) except for the mere name, the meaning is the same. Since the basic nature free from the reference points of the four extremes, that is, Dzogchen (the luminosity that is to be personally experienced) is not at all like that, it is important to rely on the correct path and teacher. Therefore, you may pronounce 'illusionlike,' 'nonentity,' 'freedom from reference points,' and the like as mere verbiage, but this is of no benefit whatsoever, if you do not know the [actual] way of being of the Tathāgata’s emptiness (which surpasses the limited [kinds of] emptiness [asserted] by the tīrthikas) through the decisive certainty that is induced by reasoning."

Chögyal Namkhai Norbu states:  

"Madhyamaka explains with the four 'beyond concepts,' which are that something neither exists, nor does not exist, nor both exists and does not exist, nor is beyond both existing and not existing together. These are the four possibilities. What remains? Nothing. Although we are working only in an intellectual way, this can be considered the ultimate conclusion in Madhyamaka. As an analytical method, this is also correct for Dzogchen. Nagarjuna's reasoning is supreme."

and,

"That view established intellectually we need to establish consciously in dependence upon one’s capacity of knowledge and on convention. The way of establishing that is the system of Prasanga Madhyamaka commented upon by the great being Nāgārjuna and his followers. There is no system of view better than that."

From Jigme Lingpa:

"I myself argue ‘To comprehend the meaning of the non-arising baseless, rootless dharmakāya, although reaching and the way of reaching this present conclusion 'Since I have no thesis, I alone am without a fault', as in the Prasanga Madhyamaka system, is not established by an intellectual consideration such as a belief to which one adheres, but is reached by seeing the meaning of ultimate reality of the natural great completion."

Chokyi Dragpa states:  

"On the path of trekchö, all the rigidity of mind's clinging to an "I" where there is no "I", and a self where there is no self, is cut through with Madhyamika Prasangika reasoning and the resulting conviction that an "I" or a "self" does not exist. Then, by examining where mind arises, dwells and ceases, you become certain of the absence of any true reality."

Again from Khenpo Tsultrim Gyamtso:  

"The great scholar and master, Mipham Chokle Namgyal, said, “If one seeks to master the basic nature of alpha purity, or kadak, it is necessary to perfect one’s understanding of the view of the Prasangika, or the Consequence School.” Alpha purity describes the basic nature of mind as it is expressed in the dzogchen descriptions. If one wishes to realize dzogchen, alpha purity, or trekcho, as it is also called, then one must perfect one’s understanding of the Consequence School. That is, one must realize that the nature of reality transcends all conceptual fabrications; it cannot be described by any conceptual terms. This is the aspect of the 'expanse.' If one recognizes this, then it is easy to realize the mahamudra because, as Milarepa sang:

The view: is original wisdom which is empty. Meditation: clear light free of fixation. Conduct: continual flow without attachment. Fruition: is nakedness stripped of every stain."

From Acarya Dharmavajra Mr. Sridhar Rana:  

"The meaning of Shunyata found in Sutra, Tantra, Dzogchen, or Mahamudra is the same as the Prasangic emptiness of Chandrakirti, i. e. unfindability of any true existence or simply unfindability. Some writers of Dzogchen and Mahamudra or Tantra think that the emptiness

of Nagarjuna is different from the emptiness found in these systems. But I would like to ask them whether their emptiness is findable or unfindable; whether or not the significance of emptiness in these systems is also not the fact of unfindability- no seeing as it could also be expressed. Also some Shentong scholars seem to imply that the Shentong system is talking about a different emptiness. They say Buddha nature is not empty of qualities therefore, Buddha nature is not merely empty, it also has qualities. First of all the whole statement is irrelevant. Qualities are not the question and Buddha nature being empty of quality or not is not the issue. The Buddha nature is empty of Svabhava (real existence). Because it is empty of real existence, it has qualities. As Arya Nagarjuna has said in his Mula Madhyamika Karika: “All things are possible (including qualities) because they are empty.” Therefore the whole Shentong/ Rangtong issue is superfluous. However, in Shentong, Buddha nature is also empty and emptiness means unfindable. In short, the unfindability of any true existence is the ultimate (skt. paramartha) in Buddhism, and is diametrically opposed to the concept of a truly existing thing called Brahman, the ultimate truth in Hinduism."

from Dzogchen Ponlop Rinpoche:  

"The practice of tregcho is essential when it comes to realizing the originally pure nature of mind and phenomena. This nature is emptiness, the basic state of the Great Perfection. For this reason, a thorough grounding in the view of Madhyamaka can be a great help when receiving instructions on tregcho. With the correct view of emptiness, one can meditate effectively on original purity [ka dag]."

and a final warning from Khenpo Tsultrim Gyamtso:  

"If we still believe in existence, if we have some type of belief in something substantial, if we think that there is something that truly exists, whatever it might be, then we are said to fall into the extreme called eternalism or permanence. And if we fall into that extreme, we will not realize the true nature of reality."

--------------------------

Here is a post I made on Dzogchen and Advaita: 

Moreover, in comparing Buddhist principles such as the nature of mind, or dharmakāya with something like the Brahman of Vedanta, there are distinct differences. Brahman on the one hand is a transpersonal, ontological, truly established ultimate. Whereas dharmakāya is a buddha’s realization of śūnyatā, emptiness, brought to its full measure at the time of buddhahood, which results from the cultivation of jñāna, or a direct non-conceptual, yogic perception of emptiness. Dharmakāya is the nature of a personal continuum of mind, is epistemic and personal in nature, and is not a truly established ultimate nature.  

Emptiness is actually the antithesis of that which the puruṣa of Advaita represents; it is the absence of a svabhāva, or an essence, whereas puruṣa is actually an essence. Unlike the puruṣa of Advaita, emptiness is a non-reductive and non-affirming negation (prasajya-pratiṣedha) of all phenomena both compounded and uncompounded. Such a view is not shared by Advaita, which despite its attempts to classify its puruṣa as a subtle nature, even free of characteristics in the case of nirguṇabrahman, posits that brahman is still an essence that possesses the quality of being free of characteristics (nirguṇa), and this is the critique that Bhāviveka levels at Advaita. Bhāviveka lived during a time in India where there were many polemical debates and interactions between different traditions, addresses the distinctions in many of his expositions. This excerpt from his Tarkajvālā is especially pertinent and addresses this issue of Advaita's puruṣa possessing characteristics:  

"If it is asked what is difference between this dharmakāya and the paramātma [bdag pa dam pa] (synonymous with Brahman) asserted in such ways as nonconceptual, permanent and unchanging, that [paramātma] they explain as subtle because it possesses the quality of subtlety, is explained as gross because it possesses the quality of grossness, as unique because it possess the quality of uniqueness and as pervading near and far because it goes everywhere. The dharmakāya on the other hand is neither subtle nor gross, is not unique, is not near and is not far because it is not a possessor of said qualities and because it does not exist in a place."

Thus we see that that dharmakāya is not an entity-like "possessor" of qualities. Conversely, brahman which is an ontological entity, does possess characteristics and qualities.   

Dharmakāya is not an entity at all, but rather a generic characteristic (samanyalakṣana). As the Buddha says in the Saṃdhinirmocana, the ultimate in Buddhism is the general characteristic of the relative. The dharmakāya, as emptiness, is the conventional, generic characteristic of the mind, as it is the mind’s dharmatā of emptiness, its actual nature that is to be recognized. Liberation results from the release of the fetters that result from an ignorance of the nature of phenomena, and this is how dharmakāya is a non-reductive and insubstantial nature.  

The differentiation of brahman as an entity versus dharmakāya as a generic characteristic is enough to demonstrate the salient contrasting aspects of these principles. Dharmakāya is an epistemological discovery about the nature of phenomena, that phenomena lack an essential nature or svabhāva. Alternatively, brahman is an ultimate ontological nature unto itself. Dharmakāya means we realize that entities such as brahman are impossibilities, as Sthiramati explains, entities in general are untenable:  

"The Buddha is the dharmakāya. Since the dharmakāya is emptiness, because there are not only no imputable personal entities in emptiness, there are also no imputable phenomenal entities, there are therefore no entities at all."

Here is another succinct and pertinent excerpt from the Tarkajvālā, regarding the difference between the view of the buddhadharma and tīrthika (non-Buddhist) systems:  

"Since [the tīrthika position of] self, permanence, all pervasivness and oneness contradict their opposite, [the Buddhist position of] no-self, impermanence, non-pervasiveness and multiplicity, they are completely different."

Advaita posits a nondual, singular, ultimate puruṣa, whereas the Buddhist view involves recognition that the diversity of countless and discrete, conventional individual entities are themselves endowed with a conventional nondual essence because they ultimately do not have an essence at all.  

The first verse of the Rig pa khyu byug points this:  

"The primal nature (prakṛti) of diversity is nondual." 

You cannot have a nondual nature of diversity if there is no diversity. Advaita Vedanta states that only the singular puruṣa is nondual in nature.  

Further, the puruṣa of Advaita involves an ontological nonduality. An ontological nonduality (advaita) is monistic in nature. Buddhism champions a different type of nonduality (advāya), which is epistemic instead of ontological.  

An ontological nonduality is where everything is reduced to a single substance that exists alone by itself, which is the definition of monism. For example if subject and object were merged and we then held a view that the union of the two as a single X is truly substantial and valid.  

On the other hand, an epistemological nonduality is simply a recognition that the nature of phenomena is free from the dual extremes of existence and nonexistence, hence "nondual". This is a non-reductive nonduality because it does not leave anything in its wake, there is no X left over once the nature of phenomena is recognized. Hence the iconic “emptiness of emptiness.”  

In epistemic nonduality the nature of a conditioned phenomenon (dharma) and its nonarisen nature (dharmatā) are ultimately neither the same nor different, hence they are "nondual", because the misconception of a conditioned entity is a byproduct of ignorance, and therefore said entity has never truly come into existence in the first place. This means that the allegedly conditioned entity has truly been unconditioned from the very beginning. And to realize this fact only requires a cessation of cause for the arising of the misconception of a conditioned entity, i.e., a cessation of ignorance. If dharmins and dharmatā were not nondual then it would be impossible to recognize the unborn nature of phenomena because that nature would be rendered another conditioned entity.  

The implications of this means that buddhadharma in general are not actually proposing a real dharmatā or ultimate nature. Which directly contradicts a teaching like Advaita Vedanta.  

Further, Advaita Vedanta is rooted in a Sāṃkhya worldview, which differs from the Abhidharma framework that Buddhism is based on, that right there creates a firm distinction in the overall way these two systems function and view the world.  

However beyond the fact that Advaita Vedanta is a sanatanadharmic view as opposed to buddhadharma, according to Buddhist systems such as Dzogchen, Advaita is a false view that is incapable of producing liberation as defined by buddhadharma in general. The *Rigpa Rangshar* for example lists Advaita Vedanta under various wrong views, and even mentions Ādi Śaṅkarācārya by name in addressing Advaita.  

For other refutations of Advaita Vedanta you can read Śāntarakṣita‘s Tattvasaṃgraha, or Bhāviveka’s Tarkajvālā, which are two main sūtrayāna level writings which dedicate some attention to contrasting these systems. One might object and say during the time of Buddha Śākyamuni there was no Advaita Vedanta so the Buddha never addressed Advaita directly, however Sāṃkhya yoga was around during the Buddha’s time, and given the Buddha separated and distinguished his dharma from these other views such as Sāṃkhya, and Sāṃkhya is the underlying worldview that Advaita is based on, we can know (or confidently infer) that the Buddha would have also objected to Advaita Vedanta.  

Sometimes people balk at these comparisons and say this is too much of a generalization, Advaita Vedanta is a variegated system, there is Sṛīṣṭīdṛīṣṭivāda, Dṛīṣṭisṛīṣṭīvāda, Māyāvāda or Vivartavāda and Ajātivāda, and of course that is fair, buddhadharma is the same way, however ultimately, just as it is the case with Buddhism, despite these diverse subsystems, the underlying framework is in essence ubiquitous and uniform. We do not deviate from that framework despite the presence of varying methodologies or views within the system, and Advaita is no different. Even the much vaunted Ajātivāda which essentially an Advaita rendition of nonarising which cribs the Buddhist notion of nonarising, anutpāda, does not escape the consequences and implications of Advaita’s eternalist view. And for this reason buddhadharma would also state that Ajātivāda is incompatible with its view.  

We can look to the Madhyamakālaṃkāra for the buddhist refutation of Advaita’s Ajātivāda:  

"Therefore, the tathāgatas have said 'all phenomena do not arise' because this conforms with the ultimate. This "ultimate" in reality, is free from all proliferation. Because there is no arising and so on, nonarising and so on isn't possible, because its entity has been negated." 

The above excerpt also exemplifies why emptiness is itself empty, and why emptiness is non-reductive. Advaita Vedanta cannot justifiably make the same claim about its puruṣa.  

Are they similar in some ways? Sure. Is there benefit to be derived from understanding Advaita Vedanta on its own terms? Certainly. Can a practitioner of Buddhism potentially understand Buddhism better by understanding the views and nuances of Advaita Vedanta? Absolutely. My own teacher studied Advaita Vedanta systematically for this express purpose. But at the end of the day they are two different systems, with different bases, paths and results.