They are definitely not the same thing, far from it:
Minimal resolution usually refers to surfaces, and every surface has one: you can get it from an arbitrary resolution by consecutively blowing down $(-1)$-curves. In general, one usually speaks about minimal models.
Crepant resolution exists in every dimension, but not for everything. In fact, admitting a crepant resolution is a very special property of the variety. In particular, it means that it has (strictly) canonical singularities, that is, as soon as it has either worse or better (say $\mathbb Q$-factorial terminal, but not smooth) singularities, then there is no crepant resolution.
However, a crepant resolution of a surface is necessarily minimal.
EDIT: Added "$\mathbb Q$-factorial" to rule out small resolutions following Karl's comment.