They both can be controlled wirelessly, both will use the same interface. So aside from the power differences (both in terms of consumption and performance), provided that they are both powerful enough to play back whatever, and provided that the 24fps bug isn't present with the Mac mini (or you don't notice it), there shouldn't really be any functional differences.
EDIT: Just noticed the title said "server". In that case, which one is better really depends on the model, however any dual core model from 2011 onwards should be roughly comparable to a Haswell NUC (slightly quicker than a NUC), while the quad core models will of course be far more powerful.