Hofstadter says that proofs and derivations are different:
A proof is something informal, or in other words a product of normal thought, written in a human language, for human consumption. All sorts of complex features of thought may be used in proofs, and, though they may "feel right", one may wonder if they can be defended logically. That is really what formalization is for. A derivation is an artificial counterpart of a proof, and its purpose is to reach the same goal but via a logical structure whose methods are not only all explicit, but also very simple.
But Wikipedia points them to the same place!