Not that much difference in the long run. When I use the word function I generally mean that a point maps to a single point. So, if a point might map to several points, I am not going to use that word, more likely mapping or transformation. In a recent article I had one of these, each point went to several points, and each point in the image probably had several pre-images, so I emphasized, in a traditional phrase, that the mapping was "many-to-many." Now, both primage and image were equivalence classes under a weaker equivalence, so the mapping did induce a function from "genus" to "genus," but was not well-defined on the level of isometry classes of quadratic forms.
Anyway, if a point goes to only a single point, you are allowed to call it a function.
EDIT: I see, you have finished college and are just asking about preferences. I've got to think about popularity in English... Function is used for $\mathbb C \mapsto \mathbb C,$ also maps from any smooth manifold to the reals. I might use function for almost any map into $\mathbb R^n$ from almost anything, but would be less likely to use function for a mapping between two other manifolds. Various kinds of mappings in algebra are unlikely to be called function.