In what ways are the Advaita concept of 'Nirguna Brahman' and the Buddhist concept of Buddha Nature similar and different? In what ways are the Advaita concept of 'Nirguna Brahman' similar, and/or different, to the Buddhist concept of Buddha Nature?

This is a good comparison, that is where the doubt arose:
Nonduality in Buddhism and Advaita Vedanta

Maybe there is a similarity...?
One short paper that compares and contrasts Vedanta and Buddhism is Acharya Mahayogi Sridhar Rana Rinpoche's essay &quot;Madhyamika Buddhism vis-à-vis Hindu Vedanta&quot;.

When we compare the Advaita Vedanta, especially as interpreted by Shankara and Madhyamika, whether be it the Svatantric from of Bhabya or Prasangic form of Candrakirti, the sharing of the same language, culture and analogies while talking about two different paradigms becomes obvious. Because of the use of the same language structure (be it Pali or Sanskrit) and the same analogies to express two different paradigms many Vedantins or scholars of Buddhism with Vedantic backgrounds have been fooled into thinking Buddhist Madhyamika is a re-interpretation of Hindu Vedanta.
In fact, the Buddha, after long years of Brahmanic as well as Sramanic meditation, found the concept of Brahma (an ultimately real, unchanging, eternal substratum to this ephemeral transient world) not only inadequate to solve the basic issue of humanity- i. e. sorrow (dukha) and questioned the very existence of such an eternal substratum; but also declared that a search for such an imagined (Skt. Parikalpita Atman) Brahman was a form of escapism and therefore not really spiritual [...]

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Since the concept of Brahman, the truly existent (Skt. paramartha sat) is the very foundation of Hinduism (as a matter of fact some form of an eternal ultimate reality whether it is called God or Nature is the basis of all other religious systems); when Buddhism denies such an ultimate reality (Skt. paramartha satta) in any form, it cuts at the very jugular veins of Hinduism.

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Many Hindu scholars believe that without an ultimate eternal reality then there can be no liberation from the changing, transient samsara; therefore even though the Buddha denied the ultimate reality, he could have meant only conceptually really existing reality, no the eternal ultimate reality which is beyond concepts. Otherwise there cannot be liberation. The fault with this kind of thinking is that it is measuring the thesis (which is no thesis) of the Buddha (or interpreting the Buddha) from within the Hindu paradigm. Remaining within the Hindu paradigm, an eternal ultimate reality is a necessity (a necessary dead end as the Buddha saw it) for the soteriological purpose, i. e. for liberation. Since according to the Buddha there is no Brahman- such a concept being merely an acquired fabrication (skt: parikalpana) learned from wrong (skt: mithya) scriptures, hankering after, searching for such a Brahman is necessary a dead end which leads nowhere, let alone liberation. The Buddhist paradigm if understood correctly, does not require an eternal something or other for liberation. In Buddhism liberation is not realizing such a ground but rather a letting go of all grounds, i. e. realizing groundless.

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But even though Sankara copied the use of these words from Buddhism and also copied many other conceptual words from Nagarjuna to elucidate his Vedantic paradigm, the paradigm that he tries to clarify with these words different. [...] In the Vedantic context, the relative truth (Skt. samvritti satya) is that this samsara is an illusion and the ultimate truth (skt. paramartha satya) is that there is an ultimately existing thing (skt. paramartha satta) transcending/ immanent in this world. The relative truth will vanish like a mist and the both transcendent and immanent Brahman will appear as the only Truth, the world being false. To sum it up, the Vedantic ultimate truth is the existence of an ultimate existence or ultimate reality. Reality here is used as something which exists (skt. satta). However, the Buddhist ultimate truth is the absence of any such satta i. e. ultimately existing thing or ultimate reality.