3.5nd dnd – Does the Duergar's extension and invisibility abilities resemble a spell or psi? And is it important?

It's not either.

The chapter on playable races tells us to consult the chapter on monsters to see how the psi abilities work. A box at the bottom of page 184 of the Monster chapter indicates that the creatures of the XPH are psionic versions of ordinary creatures *. So, you can use whatever you choose.

The way we handled this when we were playing / I was DM in a club was to decide beforehand whether the campaign or adventure was using psionic or not, and then we were using only psi or normal versions. Mixing them up did not seem like a good idea, but it is true that we did not find any rules preventing this.

* If you want to nibble, it only talks about the creatures in the Monster chapter, but there's nothing about it in the playable breeds chapter, so that seems like a pretty reasonable extrapolation.