"Emptiness" in Buddhism is analogous to "maya" in Advaita Vedanta. Both point to the (relative) unreality of phenomenal existence, when compared to the (utter) reality of the un-conditioned. Thus, neither "emptiness" nor "maya" is sheer non-existence, as a "maya-vada" interpretation of the Buddhist Heart Sutra demonstrates:
Thus have I once heard:
The Blessed One [i.e., the Buddha] was staying in Rajagrha at Vulture Peak along with a great community of monks and great community of bodhisattvas, and at that time, the Blessed One fully entered the meditative concentration on the varieties of phenomena called the Appearance of the Profound. At that very time as well, holy Avalokitsevara, the bodhisattva, the great being, beheld the practice itself of the profound perfection of wisdom, and he even saw the five aggregates as an illusion of inherent nature. Thereupon, through the Buddha's inspiration, the venerable Sariputra spoke to holy Avalokitsevara, the bodhisattva, the great being, and said, "Any noble son who wishes to engage in the practice of the profound perfection of wisdom should train in what way?"
When this had been said, holy Avalokitsevara, the bodhisattva, the great being, spoke to venerable Sariputra and said, "Sariputra, any noble sons or daughters who wish to practice the perfection of wisdom should see this way: they should see insightfully, correctly, and repeatedly that even the five aggregates are an illusion of inherent nature. Form is illusion, illusion is form, Illusion is not other than form, form is also not other than illusion. Likewise, sensation, discrimination, conditioning, and awareness are illusion. In this way, Sariputra, all things are illusion; they are without defining characteristics; they are not born, they do not cease, they are not defiled, they are not undefiled. They have no increase, they have no decrease.
"Therefore, Sariputra, in illusion there is no form, no sensation, no discrimination, no conditioning, and no awareness. There is no eye, no ear, no nose, no tongue, no body, no mind. There is no form, no sound, no smell, no taste, no texture, no phenomenon. There is no eye-element and so on up to no mind-element and also up to no element of mental awareness. There is no ignorance and no elimination of ignorance and so on up to no aging and death and no elimination of aging and death. Likewise, there is no suffering, origin, cessation, or path; there is no wisdom, no attainment, and even no non-attainment.
"Therefore, Sariputra, since the bodhisattvas have no obtainments, they abide relying on the perfection of wisdom. Having no defilements in their minds, they have no fear, and passing completely beyond error, they reach nirvana. Likewise, all the Buddhas abiding in the three times clearly and completely awaken to unexcelled, authentic, and complete awakening in dependence upon the perfection of wisdom.
"Therefore, one should know that the mantra of the perfection of wisdom - the mantra of great knowledge, the precious mantra, the unexcelled mantra, the mantra equal to the unequalled, the mantra that quells all suffering - is true because it is not deceptive. The mantra of the perfection of wisdom is proclaimed:
tadyatha - gate gate paragate parasamgate bodhi svaha!
Sariputra, a bodhisattva, a great being, should train in the profound perfection of wisdom in that way."
Thereupon, the Blessed One arose for that meditative concentration, and he commended holy Avalokitsevara, the bodhisattva, the great being. "Excellent!" he said. "Excellent! Excellent! Noble child, it is just so. Noble child, it is just so. One should practice the profound perfection of wisdom in the manner that you have revealed - the Tathagatas rejoice!" This is what the Blessed One said.
Thereupon, the venerable Sariputra, the holy Avalokitsevara, the bodhisattva, the great being, and that entire assembly along with the world of gods, humans, asuras, and gandharvas, all rejoiced and highly praised what the Blessed One had said.
A few notes:
1. "An illusion of inherent nature" replaces the original "empty of inherent nature". In Buddhism, to be empty of inherent nature is to be (1) impermanent; (2) unsatisfactory; and (3) without an unchanging "core". In Advaita Vedanta, these three characteristics can also be used to describe "the power of maya", or "illusion"; and the phrase "an illusion of inherent nature" could be understood advaiticly as describing how "inherent nature", or the un-conditioned reality, actually serves as the basis for the appearance of illusion. "An illusion of inherent nature" refers to the illusion of (or "produced by" and "not different from, nor the same as") inherent nature.
2. "Form is illusion, illusion is form" replaces the original "Form is empty, emptiness is form". By "form" is meant specifically matter-and-energy of the kind known to modern Western science; but "form" is also short-hand for all of phenomenal reality.
3. "Illusion is not other than form, form is also not other than illusion" replaces "Emptiness is not other than form, form is also not other than emptiness". In Buddhism, the human body, the form-body, is the site of enlightenment. No human body, no form-body, means no illusion, no emptiness, and thus no enlightenment.
4. Enlightenment, liberation, salvation require illusion, require emptiness. If things were not impermanent, no change would be possible. If things gave perfect and total satisfactions, humans would not seek the un-conditioned. If things possessed an unchanging "core", humans would be fulfilled by grasping such things. However, because those three characteristics of the phenomenal worlds are true, the realization of nirvana or Brahman is possible. In Buddhism, the existence of emptiness means the creation, existence, and further-existence of the worlds. Emptiness makes possible freedom from suffering. Likewise, in Advaita, illusion means that things can change, develop, evolve. Without illusion, there is no possibility for growth, for learning. Because illusion is, the world can be enjoyed. The very possibility of enlightenment, liberation, or salvation is only due to the existence of illusion.