Instead of the word "variant," chess problemists use the term "twin" (at least if there is just one variant, which is the most common situation). I don't think that there are any strict rules about what constitutes a "twin," but in general, the smaller the change, and the more tightly related the two solutions are, the more aesthetic the twin is considered to be.
Below is a problem (and its twin) composed by Aleksey Gasparyan and Ashot Egyazaryan (StrateGems 47, July-September 2009). White to move and mate in two. It is interesting to see why the key move for one position doesn't work in the other, and vice versa.