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Well some say the truth can be interpreted in different ways but if advaita says self exists and buddhism denies it there is a big problem you can't make them compatible with one another.
Actually when asked point-blank if there is or is not a self, the Buddha refused to answer, on the ground that either yes or no would lead to extremes of wrong views not conducive to the progress on the Path, or in worst cases, completely block the path to awakening:
"'Everything exists': That is one extreme. 'Everything doesn't exist': That is a second extreme. Avoiding these two extremes, the Tathagata teaches the Dhamma via the middle: From ignorance as a requisite condition come fabrications. From fabrications as a requisite condition comes consciousness. From consciousness as a requisite condition comes name-&-form. From name-&-form as a requisite condition come the six sense media. From the six sense media as a requisite condition comes contact. From contact as a requisite condition comes feeling. From feeling as a requisite condition comes craving. From craving as a requisite condition comes clinging/sustenance. From clinging/sustenance as a requisite condition comes becoming. From becoming as a requisite condition comes birth. From birth as a requisite condition, then aging & death, sorrow, lamentation, pain, distress, & despair come into play. Such is the origination of this entire mass of stress & suffering." ~~ SN 12.15 ~~
Also see Ven. Thanissaro's great essay "Questions of Skill"
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There are different types of enlightenment. All are conditional except the one attained by Buddha. One may spend eons under bliss thinking it is nibbana but it is not unless sabbe Dhamma Anatta is realised. Philosophy of Anatta is at the core of Buddhism and realisation of it is called Nirvana... any thing else no matter how blissful wont last forever. For example one may realise I am earth and enjoy the nature of earth but it won’t last forever because it is conditional and one day earth will be all covered up by water... I do not know the nature of Adi Sankaraharyas ‘enlightenment’ but if it is not based on the idea of Anatta, it won’t last forever...suffering will return.
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I find this answer to be comprehensive:
“5 Reasons why sabbe dhamma anatta does not apply to Nirvana or the Buddha
Here are 5 reasons why sabbe dhamma anatta does not apply to the Buddha or his enlightenment – because the Buddha himself said that:
So All “things” are not self because they are “things” and are not you. All things means “all of the phenomena that you perceive through your senses” – are perceptions and are not the perceiver – all your perceptions are not you.”
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You're absolutely right that they are only superficially appearing similar, but when you dive deep, they are completely different.
This has been discussed in numerous answers, which I will not repeat here - this answer, this answer, this answer, this answer, this answer.
The original teachings of the Buddha are systematic (the four noble truths), empirical (the three marks of existence), soteriological (the Noble Eightfold Path) and avoids intellectual gymnastics and obscure mysticism (Parable of the Poisoned Arrow, Acintita Sutta, Sabba Sutta), while Advaita is metaphysical and based on logic, philosophy and intellectual gymnastics.
For e.g. the Eternal Self as the Universal Consciousness or Cosmic Consciousness that is standalone, indivisible, eternal and witnesses through all living beings is a very elegant idea in Advaita Vedanta.
But the Buddha's empirical observation and realization is that consciousness is dependently originated on the six sensory media, and it is conditioned and impermanent (see MN 38). Even the mental idea of the self is also dependently originated, conditioned and impermanent. This can be realized generally through the Noble Eightfold Path, and specifically through vipassana.
So, which is true? If you dive deeper, you can see for yourself which is true. One is a very elegant intellectual model, while the other is empirically verifiable.