I like to think of group (co)homology topologically, so I would say the integral (co)homological dimension of G is the integral (co)homological dimension of BG. Thinking this way there are lots of geometric examples in which the integral homological dimension is less than the integral cohomological dimension: for example, non-orientable surfaces are classifying spaces for their fundamental groups. Their integral homological dimension is 1, and their integral cohomological dimension is 2.
Of course for other coefficient systems, this will no longer be the case (as Tom points out in his comment below). Note, for example, that the rational cohomological dimension of a non-orientable surface is 1.
[Okay, I'm going to admit some confusion in regards to the comments on the original question. Am I thinking of the flat or the injective dimension here, when I take homology of BG? From the comments, it sounds like this must correspond to the flat dimension? I don't have Brown's book in front of me to un-confuse myself...]