There really is no comparison between those machines:
- the C=64 is a slow 8-bit computer with a chipset (VIC-II and SID) intended for 2D arcade style gaming with very good (for the time) sound
- the Atari ST is a cheaply built, very fast (for it's time and cost) general purpose computer with no stronger emphasize on gaming than on word processing or MIDI controlling
The main disadvantage of the ST if you are on the wrong side of the pond it that many games have a default (some even fixed) 50Hz screen refresh rate because the vast majority of them was written in Europe (mainly in the UK, France, Germany and Spain) and the best quality video output for gaming (the only available one on pre-FM models) is RGB, so west of the pond one will need e.g. a NEC 1970VX (that despite its official specifications will display the ST's low resolutions) and an adapter cable while east of it a SCART cable is easily available and good CRTs are a dime a dozen (some cheap modern TVs will have issues with the ST's video signal, though).
Depending on what style of games one prefers, either machine will be "better". The ST has it's strenghts in classic text adventures (The Pawn e.g.), early 3D/pseudo 3D games (Starglider, Carrier Command, Vroom), RPGs (Dungeon Master), strategy (Populous) and puzzle games (Oxyd), while it's 2D titles are a mixed bag (there's a ton of lackluster titles and only few good ones, often by the Bitmap Brothers, Gremlin Graphics or Thalion). For 2D titles, the Amiga is clearly the better choice.
The C=64 will struggle with even primitive 3D (e.g. Elite C=64 vs. the ST version), but has a ton of excellent arcade style 2D games (Mayhem in Monsterland, Katakis, Pitstop II, Great Giana Sisters, etc.).