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.
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 terminal, but not smooth) singularities, then there is no crepant resolution.
However, a crepant resolution of a surface is necessarily minimal.