How is Nagarjunas 'Shunyata' different from Buddhas 'Anatta'?

According to the Pali Theravada suttas (e.g. SN 35.85), the Buddha taught 'sunnata' means 'empty of self & anything pertaining to self'. This includes Nibbana because Nibbana is one of the sense objects (Ud 8.1; ayatana; sense base) included with 'mind objects' in SN 35.85 and because the suttas also say all things are anatta (Dhp 279).

Most importantly, Nibbana is regarded a thing (dhamma) or element (dhatu; MN 115) in the Pali suttas. However, per MN 115 & many other suttas, Nibbana is the unconditioned thing (asankhata dhatu), as follows:

There are these two elements: Dve imā, ānanda, dhātuyo—

the conditioned element and the unconditioned element. saṅkhatādhātu, asaṅkhatādhātu.

When a mendicant knows and sees these two elements, Imā kho, ānanda, dve dhātuyo yato jānāti passati—

they’re qualified to be called ‘skilled in the elements’.” ettāvatāpi kho, ānanda, ‘dhātukusalo bhikkhū’ti alaṁvacanāyā”ti.

MN 115

It follows the Buddha, according to the Pali suttas, never ever not once never ever taught: all phenomena are conditional and empty of any self essence. While Nibbana is empty of any self essence, Nibbana is not conditional.

In summary, it is most essential to understand the verses below:

277. "All conditioned things [which excludes Nibbana] are impermanent" — when one sees this with wisdom, one turns away from suffering. This is the path to purification.

278. "All conditioned things [which excludes Nibbana] are unsatisfactory" — when one sees this with wisdom, one turns away from suffering. This is the path to purification.

279. "All things [which includes Nibbana] are not-self" — when one sees this with wisdom, one turns away from suffering. This is the path to purification.

Dhp 279

Also, in the Theravada Abhidhamma, the concept "sabhava" does not mean "own-being". Instead, it means "inherent/own nature", such as in the description below from the Buddha's words:

"And why do you call it 'form'? Because it is afflicted, thus it is called 'form.' Afflicted with what? With cold & heat & hunger & thirst, with the touch of flies, mosquitoes, wind, sun, & reptiles. Because it is afflicted, it is called form.

"And why do you call it 'feeling'? Because it feels, thus it is called 'feeling.' What does it feel? It feels pleasure, it feels pain, it feels neither-pleasure-nor-pain. Because it feels, it is called feeling.

"And why do you call it 'perception'? Because it perceives, thus it is called 'perception.' What does it perceive? It perceives blue, it perceives yellow, it perceives red, it perceives white. Because it perceives, it is called perception.

"And why do you call them 'fabrications'? Because they fabricate fabricated things, thus they are called 'fabrications.' What do they fabricate as a fabricated thing? For the sake of form-ness, they fabricate form as a fabricated thing. For the sake of feeling-ness, they fabricate feeling as a fabricated thing. For the sake of perception-hood... For the sake of fabrication-hood... For the sake of consciousness-hood, they fabricate consciousness as a fabricated thing. Because they fabricate fabricated things, they are called fabrications.

"And why do you call it 'consciousness'? Because it cognizes, thus it is called consciousness. What does it cognize? It cognizes what is sour, bitter, pungent, sweet, alkaline, non-alkaline, salty, & unsalty. Because it cognizes, it is called consciousness.

SN 22.79.

While the Buddha never used the term "sabhava", the text above shows the "inherent nature" of consciousness is to cognise; the "inherent nature" of feeling is to feel; etc.

To say consciousness has no inherent nature appears to say cognition (seeing, hearing, tasting, etc) is not possible.

If consciousness did not have the inherent nature to only cognise, then consciousness would be able to speak, walk, bleed, be reincarnated, etc.

In summary, because consciousness has inherent nature it cannot have 'woke' characteristics, such as 'fluidity'. Consciousness can only be consciousness; consciousness can only cognise. If consciousness had no "sabhava", there would be no sense or life experience; instead; all would be blind, deaf, dumb, etc.

Many non-Madhyamaka sects believe that the Buddha taught that things are "empty of yourself," but are not "empty of themselves." In order to fully investigate your question, you would need to investigate the Abhidharma of a non-Madhyamaka sect. The only current sect who has an Abhidharma is the Theravada sect. If you study their Abhidhamma and their Abhidhamma treatises, you'll find that they understand "Nibbana" to be a particular discrete dhamma with its own existence that is "not empty." Furthermore, they believe that the dhammas are "not empty" because they are merely empty of "yourself." They have a concept called "sabhava" which constitutes the "intrinsic essence" or "own-being" of a dhamma (which is antithetical to Madhyamaka analysis). You can read more about the Theravadin-specific framing of "emptiness" in Abhidhammatthasaṅgaha, which is available as a PDF for-free on the Internet. It starts by outlining the Theravadin four constituents of "ultimate reality," namely form, citta, cetasikas, and Nibbana. These four things, from a traditional Theravadin POV, are not empty of themselves. They are ultimate realities. According to Venerable Nagarjuna's analysis given in the MMK, emptiness itself is the only ultimate reality, and emptiness even is empty of itself.

I didn't want to have to go into unnecessary detail for such a simple question with such a simple answer, and this will be difficult to read -- but for reference, this is from the Chinese recension of the Muladhyamakakarika with commentary by Venerable Vimalaksa (T1564). The context is a debate with a Sammatīya interlocutor framed by the verses of Venerable Nagarjuna's root text:

[Root text] If nirvāṇa exists, then nirvāṇa will be conditioned. Truly, there is not one phenomenon that is unconditioned.

[Ven Vimalākṣa] Nirvāṇa is not an existence. Why? All the many things conform to dependent origination. All are conditioned. There is not one phenomenon that is called 'the unconditioned.' Although the constant phenomena are conventionally called 'unconditioned,' via analysis we find the inconstant phenomena (to be) nonexistent. How much moreso the constant phenomena, which cannot be seen and cannot be conceived?

[Root text] If nirvāṇa exists, why is it called 'independent?' There is nothing not following from dependence that is called an existent phenomenon.

[Interlocutor] If existence is not nirvāṇa, then nonexistence must be nirvāṇa.

[Root text] Existence is not nirvāṇa, much less is it nonexistence. If nirvāṇa does not exist, how could it exist as nonexistence?

[Ven Vimalākṣa] It is because of existence that there is nonexistence. If there is no existence, how can there exist nonexistence? Like the scriptures say, former existence presently not existing is what is called 'nonexistence.' Nirvāṇa, however, is not so. How? It is not an extant phenomenon changing into the nonexistent. Because of this, nonexistence also is not nirvāṇa.

Traditional Theravadin orthodoxy disagrees with the bolded statement. They do believe that there is a "particular phenomenon" called "Nirvana" that is not empty of itself, is not empty of its own particular characteristics that are intrinsic to it.

In the suttas when Buddha speaks about Anatta (really, Anatva - as we now know from the Gandhari manuscripts) he usually goes on to explain that no (conditional) dharma is fitting to be pointed at and identified as "that's me", "that's what I'm made of", "that's mine". Wherever we look we only see physical, informational, energetic, mental, and emotional phenomena. They interact, impacting each other, combining together and splitting apart, creating latent potentials, and being created by latent potentials - but all of it is utterly impersonal. So in its basic sense Anatta simply means that our habit of pointing at some of that and saying "that's what i am" is a folly that only leads to frustration as a piece we identified with undergoes transformation or decomposition.

Now, in the few suttas where the Buddha talks about Shunya and Shunyata he uses the word in context of explaining another bad habit we have - projecting our preconceptions on whatever it is that we perceive and then assuming that our interpretations are the reality. Buddha says, that all phenomena are in fact empty (=shunya, adjective) of what we read into them. All we have is our own senses, the rest is only our interpretation. When we clearly understand that, we can "dwell in Emptiness" (=Shunyata, noun) by seeing things as they are, without projecting our prejudices on them. This is what Jidu Krishnamurti calls "freedom from the known" and what the Buddha explains with the metaphor of the sunlight not falling anywhere when the ground is removed.

Nagarjuna picks up on these themes and explains that what Buddha must have meant is that the bad habit of reification (confusing our simplistic interpretations for reality) is in fact the main driving force behind craving/attachment and therefore is the root cause of suffering. He then goes on to illustrate how every fixed entity or notion that we can take for analysis is in fact logically inconsistent under careful examination. He means that as a proof that all so-called things are only our simplistic interpretations of fluid, non-discrete, amorphous, ungraspable reality.

Nagarjuna then goes on to explain, that Dependent Origination in fact describes the process of gradual development of interpretation and onset of reification of it. He shows that all so-called "distinct things" are empty exactly because they arose as a result of dependent origination - because they are interpretations, not reality. He summarizes this in his famous assertion: Emptiness and Dependent origination describe the same principle - (dependently originated) form is empty, and emptiness is the true ground of all forms, "Form is empty and emptiness is form".

Before Nagarjuna, Anatta and Shunyata were rather different. Anatta referred to the folly of identifying with something and Shunyata referred to the folly of reifying our interpretations of phenomena.

After Nagarjuna, Anatta got subsumed in Shunyata as a more broad principle. Because all our ideas of forms or external phenomena, and our idea of self or the subject, result from Dependent Origination, both are empty. Thus, after Nagarjuna, Anatta is understood as merely a one specific case of Shunyata which has now received its long-deserved recognition as the central concept of Buddha's teaching.