How do I refute the claim that the Buddha was actually preaching Vedanta? Some people are obsessed with making a great personality and revolutionary like the Buddha a follower of their own faith by making baseless, historically inaccurate and factually incorrect claims. One such claim is that Buddha was a Hindu Yogi who actually preached Vedanta. How do I refute this claim?
This is a modified version of the answer that I posted on Hinduism.SE comparing the Buddhist Madhyamaka (by Nagarjuna) and the Hindu Advaita Vedanta (by Adi Shankara). Usually, the emptiness in Madhyamaka is equated to Nirguna Brahman (formless attributeless God) in Advaita by Hindu scholars like Prof. Chandradhar Sharma. Prof. Sharma also equated the Eternal Buddha concept (based on the Buddha's Dharma body) to God. I modified this answer for posting here.

First: What did Adi Shankaracharya teach?

We can see this from Adi Shankaracharya's own compositions:


  Brahman is real, the universe is an illusion. The jiva is Brahman
  itself and not different. (Brahmajnanavalimala 20)
  
  This Atman is a self-cognised entity because It is cognised by Itself.
  Hence the individual soul is itself and directly the Supreme Brahman,
  and nothing else. (Vivekachudamani 216)
  
  There exists no other material cause of this phenomenal universe
  except Brahman. Hence this whole universe is but Brahman and nothing
  else. (Aparokshanubhuti 45)
  
  The pot, wall, etc., are all nothing but clay. Likewise, the entire
  universe is nothing but Brahman. (Brahmajnanavalimala 19)
  
  Brahman is Existence, Knowledge, Infinity, pure, supreme,
  self-existent, eternal and indivisible Bliss, not different (in
  reality) from the individual soul, and devoid of interior or exterior.
  It is (ever) triumphant. It is this Supreme Oneness which alone is
  real, since there is nothing else but the Self. Verily, there remains
  no other independent entity in the state of realisation of the highest
  Truth. All this universe which through ignorance appears as of
  diverse forms, is nothing else but Brahman which is absolutely free
  from all the limitations of human thought. A jar, though a
  modification of clay, is not different from it; everywhere the jar is
  essentially the same as the clay. Why then call it a jar ? It is
  fictitious, a fancied name merely. None can demonstrate that the
  essence of a jar is something other than the clay (of which it is
  made). Hence the jar is merely imagined (as separate) through
  delusion, and the component clay alone is the abiding reality in
  respect of it. Similarly, the whole universe, being the effect of
  the real Brahman, is in reality nothing but Brahman. Its essence is
  That, and it does not exist apart from It. He who says it does is
  still under delusion - he babbles like one asleep. This universe is
  verily Brahman - such is the august pronouncement of the Atharva Veda.
  Therefore this universe is nothing but Brahman, for that which is
  superimposed (on something) has no separate existence from its
  substratum.  (Vivekachudamani 225-231)
  
  Therefore the universe does not exist apart from the Supreme Self; and
  the perception of its separateness is false like the qualities (of
  blueness etc., in the sky). Has a superimposed attribute any meaning
  apart from its substratum? It is the substratum which appears like
  that through delusion.  (Vivekachudamani 235)
  
  Becoming thyself the self-effulgent Brahman, the substratum of all
  phenomena - as that Reality give up both the macrocosm and the
  microcosm, like two filthy receptacles. (Vivekachudamani 289)


The above quotes clearly show the teaching of eternalism.

Brahman is eternal, infinite, self-existent and transcendental absolute reality. Brahman is the substratum or foundation of the universe. Brahman is the material cause of the universe. Brahman is the only thing that is permanent and ultimately real, while the universe is ultimately an illusion.

The individual soul, which is a self-cognized entity, is ultimately the same as Brahman.

While Brahman is without qualities, it appears as the universe and individuals with qualities, through delusion.

This summarizes the nature of reality according to Advaita Vedanta.

Second: What did the Buddha teach?

We can see this in the Buddha's own words (Buddhavacana), from the Sutta Pitaka in Pali, which is included in the Mahayana Agamas and the Tibetan Kangyur.

It is widely speculated that some Buddhist schools teach annihilationism or nihilism, but this is not true. It is only non-Buddhists who have this wrong impression. The Buddha and 99.9% of Buddhist schools teach the middle way between eternalism and annihilationism. This "middle way" teaching is the unique feature of the Buddha's teachings.

The Buddha rubbished the notion that there is no self of any kind:


  “So, brahmin, when there is the element of endeavoring, endeavoring
  beings are clearly discerned; of such beings, this is the self-doer,
  this, the other-doer. I have not, brahmin, seen or heard such a
  doctrine, such a view as yours. How, indeed, could one — moving
  forward by himself, moving back by himself — say ‘There is no
  self-doer, there is no other-doer’?” - Attakari Sutta


However, the Buddha was very clear in the Ananda Sutta (SN44.10) that rejecting both eternalism and annihilationism,  he teaches that "all phenomena is not self", which means that there is no permanent or eternal standalone entity of a self in all phenomena (sabbe dhamma anatta).


  "Ananda, if I — being asked by Vacchagotta the wanderer if there is a
  self — were to answer that there is a self, that would be conforming
  with those brahmins &amp; contemplatives who are exponents of eternalism
  [the view that there is an eternal, unchanging soul]. If I — being
  asked by Vacchagotta the wanderer if there is no self — were to answer
  that there is no self, that would be conforming with those brahmins &amp;
  contemplatives who are exponents of annihilationism [the view that
  death is the annihilation of consciousness]. If I — being asked by
  Vacchagotta the wanderer if there is a self — were to answer that
  there is a self, would that be in keeping with the arising of
  knowledge that all phenomena are not-self?"
  
  "No, lord."
  
  "And if I — being asked by Vacchagotta the wanderer if there is no
  self — were to answer that there is no self, the bewildered
  Vacchagotta would become even more bewildered: 'Does the self I used
  to have now not exist?'"


The Buddha instead taught that the self is not eternal and not non-existent, but is dependently originated (Pratītyasamutpāda):


  When the Buddha was asked by the naked ascetic Kassapa whether
  suffering was of one's own making or of another's or both or neither,
  the Buddha replied "Do not put it like that." When asked whether there
  was no suffering or whether the Buddha neither knew nor saw it, the
  Buddha replied that there was, and that he both knew and saw it. He
  then said "Kassapa, if one asserts that 'He who makes (it) feels (it):
  being one existent from the beginning, his suffering is of his own
  making,' then one arrives at eternalism. But if one asserts that one
  makes (it), another feels (it); being one existent crushed out by
  feeling, his suffering is of another's making,' then one arrives at
  annihilationism. Instead of resorting to either extreme a Tathaagata
  teaches the Dhamma by the middle way (by dependent origination)" -
  summarized here from Acela Sutta


A summary of dependent origination is that the self arises dependent on the inter-working of the five aggregates of form, feeling, perception, mental fabrications, and consciousness. Please see this answer for further elaboration, including the lute analogy.

He taught that both the views of "I have a self" and "I have no self" are inaccurate in the Sabbasava Sutta. He also taught that the views of one being Nirvana, one being in Nirvana, one being apart from Nirvana, calling Nirvana "mine", and delighting in Nirvana, are inaccurate in the Mulapariyaya Sutta.

The Buddha did teach emptiness, but it is restricted to the nature of the self i.e. all phenomena is empty of a self (see Shunya Sutta). The Buddha did not comment on the nature of all non-sentient things like the universe, besides noting them as being conditioned and impermanent.

The Buddha was not interested in commenting on the nature or origin of the universe, because he considered it to be unimportant to the path to liberation from suffering - see the Parable of the Poisoned Arrow (from MN63). The Buddha was not interested in metaphysical speculations. It was not because he taught the Upanishadic truths through silence, as implied by Prof. Chandradhar Sharma.

Third: What did Nagarjuna teach?

While the Buddha stopped at describing the "middle way" and emptiness with respect to the nature of the self, Nagarjuna expanded these concepts to cover the nature of the universe and all reality. He did this in his magnum opus, the Mūlamadhyamakakārikā, whose name itself means "Fundamental Verses on the Middle Way". Nagarjuna did not contradict the Buddha's teachings, but rather, expanded it.

The Wikipedia article on Madhyamaka (which contains its own citations) summarized Nagarjuna's teachings very well:


  Central to Madhyamaka philosophy is śūnyatā, "emptiness." The term
  refers to the "emptiness" of inherent existence: all phenomena are
  empty of "substance" or "essence" (Sanskrit: svabhāva) or inherent
  existence, because they are dependently co-arisen. At a conventional
  level, "things" do exist, but ultimately they are "empty" of inherent
  existence. But this "emptiness" itself is also "empty": it does not
  have an existence on its own, nor does it refer to a transcendental
  reality beyond or above phenomenal reality.
  
  Nagarjuna's critique of the notion of own-nature (svabhāva) argues
  that anything which arises according to conditions, as all phenomena
  do, can have no inherent nature, for what is depends on what
  conditions it. Moreover, if there is nothing with own-nature, there
  can be nothing with 'other-nature' (para-bhāva), i.e. something which
  is dependent for its existence and nature on something else which has
  own-nature. Furthermore, if there is neither own-nature nor
  other-nature, there cannot be anything with a true, substantial
  existent nature (bhāva). If there is no true existent, then there can
  be no non-existent (abhāva).


Rather than the annihilationism that nothing exists or the eternalism that something exists eternally, Nagarjuna taught the "middle way" that all phenomena is empty of its own "inherent existence" or "substance" or "essence" (what he called svabhāva).

If nothing has inherent substance, then nothing can depend on something else for substance, so there is no other inherent substance (para-bhāva).

Nagarjuna's unique philosophy however, is that even this emptiness is empty i.e. this emptiness does not have its own inherent substance. This means that there is no transcendental reality beyond phenomenal reality. This is what is implied by ultimate reality not being absolute reality.

A very nice and simplified explanation of the Madhyamaka emptiness can be found in Zen master Thich Nhat Hanh's writing, "The Fullness of Emptiness".

The only disagreement between Theravada and Madhyamaka is the need to expand the "middle way" and "emptiness" beyond what the Buddha taught. What is common between Theravada and Madhyamaka is that both reject eternalism and annihilationism.

Fourth: Comparing Advaita and Madhyamaka

Comparing what Adi Shankaracharya taught and what Nagarjuna taught, we come to see very clearly that both are not the same and in fact, completely incompatible:


Shankara said Brahman is eternal and absolute. Meanwhile, Nagarjuna said nothing is eternal and absolute.
Shankara's Brahman (clay analogy) implies that it is the only thing that has a true inherent substance (what Nagarjuna called svabhāva). Meanwhile, Nagarjuna says nothing has inherent substance.
Shankara's Brahman is the material cause (again, clay analogy) of the universe. Meanwhile, Nagarjuna's emptiness is not a material cause for anything including itself.
Shankara said that the universe depends on Brahman as its substratum (what Nagarjuna called para-bhāva). Meanwhile, Nagarjuna said there is no other inherent substance (para-bhāva) i.e. no substratum for anything else.
Shankara's Brahman is the Ultimate Reality that is the Transcendental Absolute Reality. Meanwhile, Nagarjuna's Ultimate Reality is an "emptiness of emptiness" that is devoid of transcendental or absolute reality.


What is common between Advaita and Madhyamaka is that both proclaim the non-dualism between the Ultimate Reality and the phenomenal reality. However, Advaita's Ultimate Reality is a Transcendental Absolute Reality, while Madhyamaka's Ultimate Reality is the "emptiness of emptiness" that is devoid of any transcendental or absolute reality.

My conclusion is that Adi Shankaracharya's Nirguna Brahman and Madhyamaka's "emptiness of emptiness" are not the same and in fact, completely incompatible.

This is supported by Banaras Hindu University Professor T. R. V. Murti's statement (quoted below) in this book chapter: Murti T.R.V. (1973) Saṁvṛti and Paramārtha in Mādhyamika and Advaita Vedānta. In: Sprung M. (eds) The Problem of Two Truths in Buddhism and Vedānta. Springer, Dordrecht


  It has been the fashion to consider that the differences between the
  Madhyamika śūnyatā and Brahman are rather superficial and even verbal,
  and that the two systems of philosophy are almost identical. At least
  Professor Radhakrishnan thinks so, and Stcherbatsky's and Dasgupta's
  views are not very different. I hold a contrary view altogether: that
  in spite of superficial similarities in form and terminology, the
  differences between them are deep and pervasive.


Fifth: Comparing Vedanta and Buddhism

All Vedanta schools accept that Atman (soul) is eternal (according to Bhagavad Gita 2.20), and Brahman is absolute and eternal. This is based on the teachings of the Upanishads.

Meanwhile, the anatman teaching subscribed by all schools of Buddhism, states that there is no eternal self (i.e. no permanent standalone eternal entity) in all phenomena.

A very apt conclusion for this comes from eminent German indologist Professor Helmuth von Glasenapp's essay "Vedanta and Buddhism: A Comparative Study":


  In the light of these researches, all attempts to give to the Atman a
  place in the Buddhist doctrine, appear to be quite antiquated. We know
  now that all Hinayana (sic) and Mahayana schools are based on the
  anatma-dharma theory. ... Nirvana being a dharma, is likewise anatta,
  just as the transitory, conditioned dharmas ... Nirvana is no
  individual entity which could act independently. For it is the basic
  idea of the entire system that all dharmas are devoid of Atman, and
  without cogent reasons we cannot assume that the Buddha himself has
  thought something different from that which since more than 2000
  years, his followers have considered to be the quintessence of their
  doctrine.


and


  Nothing shows better the great distance that separates the Vedanta and
  the teachings of the Buddha, than the fact that the two principal
  concepts of Upanishadic wisdom, Atman and Brahman, do not appear
  anywhere in the Buddhist texts, with the clear and distinct meaning of
  a "primordial ground of the world, core of existence, ens realissimum
  (true substance)," or similarly.