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.