Buddha Nature is known as "tathagatagarbha".
According to this article:
Tathagatagarbha, or Tathagata-garbha, means "womb" (garbha) of Buddha
(Tathagata). This refers to a Mahayana Buddhist doctrine that Buddha
Nature is within all beings. Because this is so, all beings may
realize enlightenment. Tathagatagarbha often is described as a seed,
embryo or potentiality within each individual to be developed.
And according to this article:
The 'tathagatagarbha' symbolizes the potential for enlightenment (a
principle) rather than a material "essence" of ultimate truth.
Buddha Nature is said to originate from the concept of the "luminous mind" in Pabhassara Sutta, which is anyway not eternal, not unconditioned and not impermanent (see this answer).
Nirguna Brahman is described in Advaita as the substratum of all phenomena in Vivekachudamani 289 and as the material cause of the phenomenal universe in Aparokshanubhuti 45.
So, while Buddha Nature is simply the potential for enlightenment (a principle), it is not the material "essence" of ultimate truth, which Nirguna Brahman is. So Buddha Nature and Nirguna Brahman are not the same.
The second concept often wrongly equated with Nirguna Brahman, is Madhyamaka's "emptiness of emptiness" or shunyata. This answer in Hinduism.SE explains why both are not the same. Basically, Nirguna Brahman is empty of qualities but not empty of substance. Meanwhile, shunyata is empty of substance and even this emptiness itself is empty of substance or existence. In other words, Nirguna Brahman is a Transcendental Absolute Reality, while shunyata is devoid of any transcendental or absolute reality.
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.
The third concept usually mistaken for Brahman is the concept of the Eternal Buddha. This comes from the notion that the Buddha's dharmakaya or Dharma Body is Eternal. This does not mean that the Buddha is an eternal being or entity. Rather, it means that the Buddha's teachings (called the Dharma or Dhamma) is eternal. This originates from the Vakkali Sutta where the Buddha told Vakkali:
"Enough, Vakkali! What is there to see in this vile body? He who sees
Dhamma, Vakkali, sees me; he who sees me sees Dhamma. Truly seeing
Dhamma, one sees me; seeing me one sees Dhamma."
And also from the Mahaparinibbana Sutta:
Then the Blessed One said to Ven. Ananda, "Now, if it occurs to any of
you — 'The teaching has lost its authority; we are without a Teacher'
— do not view it in that way. Whatever Dhamma & Vinaya I have pointed
out & formulated for you, that will be your Teacher when I am gone.
All Vedanta schools accept that Atman is eternal (according to Bhagavad Gita 2.20), and Brahman is absolute and eternal.
Meanwhile, the anatman or anatta teaching subscribed by all schools of Buddhism, states that there is no eternal self (i.e. no permanent standalone eternal entity) in all phenomena.
The eminent German indologist Professor Helmuth von Glasenapp explains aptly in his essay "Vedanta and Buddhism: A Comparative Study":
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.
For more discussion on the differences between doctrines in Hinduism and Buddhism, please see this answer.