I have to credit both games for getting me into fighters. I get the feeling that there is going to be an overwhelming majority that prefer 3rd Strike, and that’s okay. I think the comments as to why you prefer this game or that game will still be interesting.
As far as graphics, music, and overall presentation. I think both games are of the same level. They both feature highly detailed sprite work. It’s just a matter of personal taste. Though, I think Garou succeeds at having a better story, and each character has a clear back story and strong interactions with other characters that build on that story. The story in 3rd Strike was all ready told in Next Generation and 2nd impact; 3rd Strike just assumes that you know the story and proceeds right along. That said, both games feature very distinct characters whose personalities come alive the moment you see them.
Gameplay: 3rd Strike wins. Without question. Garou has great gameplay too, but to be honest the Just Defend system is a newb-friendly parry system. Guard Cancelling is fine. What I don’t like about Garou is the wakeup game being pretty shallow in comparison to 3rd strike, and that a large variety of pokes can interrupt nearly all special moves. Plus the block strings of Jenet, Kevin, and Terry taking away from strategy. 3rd Strike is more about player skill (except maybe Chung Li, but this debatable) and has an overall teaching of the fundamentals of fighters. The mind game alone is intense, leading to some spectacular battles spectators can easily get into and get overwhelmed by amazing victories and epic losses.
3rd Strike is my favorite fighter, with Garou being a very close second.