I think a very important concept in Buddhism is the middle way. From the perspective of practice, it refers to the middle path between indulgence and asceticism. From the perspective of philosophy, it refers to the middle way between eternalism and annihilationism. This theme keeps reappearing everywhere in Buddhism and seems to be unique to it. This is discussed in this answer.
Anatta is the concept where there is no absolute eternal permanent self, that is behind all phenomena. At the same time, there is a self that is not annihilated completely at death. The self is not standalone and emerges out of the interworking between form, sensation, perception, mental formations and consciousness. How the perception of self is formed is covered in dependent origination.
Anicca too has a connection to this middle way. The universe is neither eternal and absolute, nor is it an illusion. It is real, but it is always changing. However, the universe is empty as in devoid of a permanent self.
Another important middle way concept in Buddhism is that the person that is reborn is neither exactly the same nor totally unrelated from the person in the previous life. This is discussed in this answer.
In physics too, similar ideas appear in quantum mechanics, with the wave-particle duality as an example.
In Advaita Vedanta, Atman (the Self) is permanent, standalone and not different from Brahman, the ultimate eternal reality. The Atman is also considered the Eternal Witness, which extends to the idea of Eternal Consciousness, that every being has the same "I" which is the Atman, which is ultimately Brahman. The Self Inquiry method of Sri Ramana Maharshi asks "who am I?" and seeks to lead the spiritual seeker towards Self Realization, that "I" am not this temporary person (the little "I"), but rather the Eternal Consciousness (the big "I"), the Atman, who is Brahman.
This is quite incompatible with Buddhism, because the Buddha refuted not only a permanent self (in this answer), but also the idea of Eternal Consciousness. In this sutta, the Buddha rebuked Sati the monk, for holding the view of Eternal Consciousness (related to Atman), and corrects him by teaching the concept of eye-consciousness, ear-consciousness etc. According to this, consciousness is not an Eternal Witness, but rather, arises with sensation and perception. According to this sutta, the five aggregates are empty as in, devoid of a self.
Hence Advaita Vedanta is quite different and incompatible with Buddhism. However, according to this question, what is common or similar between the two is the fact that both Nirvana and Brahman are eternal, unborn and undying.