Well, I think the answer was in the question to begin with...
From what I gathered, the "Mac Mini" are relatively cheap PCs (PC stands for "Personal Computer", that's doesn't have anything to do with the manufacturer or the OS inside, damnit!).
Being cheap, you can't expect it to be a powerful beast.
Also, as pointed above, Mac machines tend to be very optimized for a lot of things that built them for professionals, and as such, it is usually at the expense of the gaming experience.
Intel HD cards are really great for all that is only image editing and such, and still relatively capable for 3D, and they are pretty cheap.
The downside of that is that they usually can't compare to NVidia cards (or other manufacturer that I never remember the name off), even more when you hit the high grade ones.
Great graphics card cost a lot, sadly, so you can't expect a $800 "mac mini" to perform as well as a $2k computer built for gaming.
That, and minecraft doesn't run super well and smooth, even on high end gaming computers. Optifine helps, if well configured, but the fact that it uses Java doesn't help at all.