The question is in the title, really.
I understand the answer in a general sense: Morphisms map objects, and functors map both objects and morphisms. So an endofunctor would map morphisms as well as objects. But that's not an intuitively satisfying answer. It seems like "part" of the endofunctor is just a morphism under a different name. I'm trying to understand this at a deeper level, and the texts I have don't really cover this question, per se.
Examples might be helpful, or more clarification on the ways they are different beyond the simple version I give above. I'm not sure how to clarify what I'm after yet, but feel free to try and pin me down to specifics in comments if it helps...