Advaya literally means "not two", it is an adjective and never referring to a thing in itself. Advaita is grammatically and semantically very different. It refers to a 'non-duality', and in nearly all usages is an abstract noun referring to a kind of thing, and further that very thing in itself.
In short, advaita means non-duality in the manner of being one single thing, it refers to a kind of monism. While with advaya there is no such implication, here "not two" or non-dual doesn't necessarily mean "one". Rather, Buddhism asserts ontological undecidability, some texts express this by talking about being beyond both dual and non-dual.
Therefore in context we see this play out for example where advaita vedanta believes in a real, truly existing state of 'non-duality', and constantly use the word advaita to indicate this.
Buddhism proper completely rejects this monism and non-duality, it recognizes that there is no state or condition that is ever truly devoid of all duality as such.
So what does advaya mean to Buddhism in context? Beyond "one", beyond "two", just the mere fact that all phenomena arise in dependence and are free from all ontological extremes.
Perhaps surprising to many Westerners, to Buddhism this is just a view and in itself has surprisingly small soteriological use: merely that by understanding that lack of awakening stems from not apprehending the non-dual nature of things, we can strive to overcome it. While emptiness and non-arising on the other hand aren't views but serve as cures for views.
Merely knowing about the view doesn't do much for one's practice and this hype about non-duality in the Western contemplative community is really an overused and overrated fad that early Western scholars largely perpetuated.
In fact, experiences of oneness and "all is one" are not considered very remarkable in Buddhism, since Buddhism isn't that interested experiences per se, but is concerned with specific non-conceptual direct perceptions and ascertaining the limits of conceptual proliferation.
The difference is significant:
In the incorrect view one pursues a false, conceptually tainted nature that is an ontological, transpersonal, homogeneous, unconditioned existent. It reduces all to a non-differentiated, single substance that is self-existing. This often manifests in practice by treating the clarity of mind as an abiding substratum that serves as an independent foundation for a witness or "higher self".
While the pursuit of Buddhadharma is epistemic, and regarding a personal, heterogeneous dharmata free from the extremes of existence and non-existence etc. It is insubstantial and a non-reductive recognition that there is nothing established in which or of which to be a part.
Since so called 'conditioned' phenomena and their non-arising natures are ultimately same nor different, these phenomena have never truly come into existence in the first place, and thus have been pure and empty from the very beginning.