[FORGET MY EARLIER ANSWER - I WAS STUPID] A fellowship usually refers to a grant made to graduate student or a senior staff member. A scholarship is any kind of financial aid to a student, usually an undergraduate student. You don't have to be in a school to have fellowship.

The funds given at the undergrat level are usu. called scholarships, while for post-grat studies the funds are fellowships.

A scholarship is a grant to a student, for whatever reason. A fellowship might be merit-based, or it could involve teaching or research. So a fellowship lies a little higher up the food chain.