For competition on TSC, the Genesis version is all you need. The disc versions are graphically enhanced and contain brand new special stages and music, but they also inexplicably have less rings in certain levels, making some ring and score records impossible. Also, special stages do not differ depending on whether Tails or Knuckles takes you there.
That said, gameplay-wise the Saturn version is superior in every way to the other two except for its crappy stage "skip" code instead of level select (it requires some stages previous to the one you want to access to load), closely followed by the PC version, which sheds some graphical effects and severely dumbs down the special stages, but has a save system (that you can abuse to make a level select :P).
EDIT: Of course, if you're playing the Saturn version on SSF, you can always make savestates. I personally have a state at the start of all seven special stages. They're arguably the best in the series.
EDIT 2: Oh, and how could I forget
this post.