Nyx is the result of a partially failed resurrection/incarnation spell meant to give Nightmare Moon her own body independent from Luna fueled by the blood of the element of Magic.
So no she is the bodily incarnation of Nightmare Moon, but NMN as part of Luna was nothing but a malady, to exist independently from Luna in her own body she also needed her own soul something NMN did not have. In that respect they can be said to be different. And while soul/no soul is kind of an absolute identity is not an cannot truly be considered to be anything but a continuity of factors. I have concluded that there is enough continuities that they must be considered to share a single identity. She goes by the name Nyx simply because she likes that name and what it represent better then Nightmare Moon.