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.

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. Actually, one could say minimal resolution in arbitrary dimension as well, but have to keep in mind that that is usually a) not actually a resolution (minimal models are not necesssarily smooth) b) not unique.

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.

As Karl shows, not every minimal resolution of surfaces is crepant, but it is easy to see that every crepant resolution of a surface is necessarily minimal.

EDIT: Added "$\mathbb Q$-factorial" to rule out small resolutions following Karl's comment.

EDIT2: Added comment about minimal resolutions in arbitrary dimension.

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.

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.