Fermat never gave a proof, only announced he had one (sounds familiar?). Euler did give a proof, which was flawed, see Franz Lemmermeyer's lecture notes, or see page 4 of David Cox's introduction.
For a discussion why a proof along the lines set out by Fermat is unlikely to work, see this MO posting.
---- trivia ----
As a curiosity, I looked up Fermat's original text (reproduced below from his collected works), written in the margin of the Arithmetica of Diophantus:
Can one find in whole numbers a square different from 25 that, when
increased by 2, becomes a cube? This would seem at first to be
difficult to discuss; and yet, I can proof by a rigorous demonstration
that 25 is the only integer square that is less than a cube by
two units. For rationals, the method of Bachet would provide an infinity
of such squares, but the theory of integer numbers, which is very
beautiful and subtle, was not known previously, neither by Bachet,
nor by any author whose work I have read.