Once as a grad student I handed in a proof to a logician. The proof was a good proof—by the standards of mathematicians (the proof was of a probabilistic fact, and I had previously published a number of peer-reviewed articles in probability theory, so I knew what I was doing). The logician absolutely hated it and did not think it was a proof.
What a logician means by a "proof" and what a mathematician means by a "proof" are different. I think the difference is roughly this: A mathematician's proof is an informal argument that there exists a logician's proof.