The Galaxy S III mini is nothing like the real Galaxy S III in both hardware and software. The Galaxy S III mini uses more cheaper, older and less powerful hardware than the Galaxy S III and not all features on the actual Galaxy S III is present on the Galaxy S III mini. That is the major downfall. It's not as good, it's slower, doesn't offer as much and etc. The only good thing about the mini is the how compact it is.
The Galaxy S III is a much better handset, you may think it's big at first, but you'll get used to it after a few days. Most people do. I even see people with small hands holding the Galaxy Note II with no problems at all and actually likes it. So the Galaxy S III is the same, since it's smaller than the Note II, it's much easier to hold. The S III is faster and comes with more features than the mini and once you get the Premium Suite update, you also get Galaxy Note II features such as multi-view which allows you to run two apps simultaneously in a split view (one app at the top and one at the bottom).