All their misunderstandings arise from that Bhagvan Buddha refused to discuss ultimate questions.+

Bhagavan Buddha: - There are only two mistakes one can make along the road to truth: not going all the way... and not starting.

The word Brahman means ultimate truth or reality which cannot be indicated by any word. The Brahman can be expressed through silence because it is beyond the experience of form, time,  and space.  Therefore, the word Brahma clearly stands for the essence of the three states, which is consciousness only. 

The final use of the pursuit of the truth is to know that the 'Self ' is the Soul, which is present in the form of consciousness.

Sage Sankara opposed the Buddhists only, who misunderstood Bhagavan Buddha and became atheists. According to Sage  Sankara, meditation always means a critical analysis of the self to get salvation from worldly tensions. Due to the eccentric ego of the then atheists, Sage  Sankara did not go beyond this since the atheists will not accept God beyond themselves. This limitation is not due to limited knowledge of Sage  Sankara but is due to the then-existing situation of the psychology of the surrounding society. Even Buddha kept silent about God because the society dealt with by Him consisted of Purvamimamsakas, who were strong atheists. Bhagavan Buddha said that everything,  including the Self, is only relatively real (Sunya). This is correct because the self is a part of the universe, which is relatively real with respect to the absolute unimaginable God. Bhagavan Buddha stopped at this point because atheists cannot realize the existence of unimaginable God indicated through His silence. 

The point of Bhagavan Buddha is that if God is non-existent, the entire creation,  including the Self is non-existent. Sage Sri, Sankara wanted to establish the existence of the Brahman. For this purpose, He made the Atman as the Brahman. 

He brought out the identity of self with the consciousness and made the Atman the Brahman. Since one will not negate the existence of the Self, he will accept the existence of the Brahman, which is the Atman or Soul, the Self. Both Buddha and Sage  Sankara kept silent about the absolutely unimaginable God. The same philosophy was dealt with by them from different angles in different situations. 

Buddha was a Gnani, but his interpreters were not. Buddha did not enter into the scriptural interpretation. So the Hindus threw him out of their religion Sage  Sankara however although he agreed on nearly all points with Bhagavan Buddha, he was a tactician and wanted to teach these truths within the Hindu fold. Hence,  he did in Rome as Rome does! He made himself outwardly appear as an orthodox Hindu and thus secured his aim.

Buddhism has failed by misunderstanding Gotama and believing that nothing is left to exist after Nirvana. What is it that sees the illusory nature of the finite ego? This is what the Buddhists need to answer and cannot on their theories. 

Only the Advaita can reply: it is the witness. The Buddhists are in error in regarding the finite ego as illusory, and as having nothing more behind it: but they would have been perfectly correct in such an outlook had they added the notion of the witness. 

How is it that Skandas come together and compose the ego? Who sees them come and go? It is the witness, the Atman, and this lack of Advaita supplies in the witness and witnessed Analysis. When they say that mind comes and goes they are forgetting that there must be another part of the mind as the consciousness which notices it and which tells them of this disappearance and appearance. All their misunderstandings arise from the fact that Bhagavan Buddha refused to discuss the ultimate questions. 

When Buddhism degenerates into Nihilism Advaitic Sages refute it (See Manduka P.281).

 The truth of a single reality within or underlying the illusory ego is all-important and without it Buddhism becomes fallacious.

Vedanta admits the transitoriness and evanescence of thoughts just like Buddhism, but not of the  Atman which observes this transitoriness and knows it. Buddhists borrowed from the Upanishads because they were Indians. The Vedantins did not need to borrow from Buddhism therefore (see P.396 v.99 of Manduka Up): ~Santthosh Kumaar