Before I say this... do NOT think I'm saying these two can't act... both were great.
Both, however, were pretty much the same in all their roles. That was ok.... because it was great.
Bogart was in more 'great' films, IMHO. But Cary... well, there was just something about that guy. Loved watching him... you couldn't help but love the guy.
I'm like Rambeau in that I think "Casablanca" Bogey's best film... BUT... "Treasure of the Sierra Madre" in nearly equal, IMHO.
My favorite Grant film is an odd choice... well, two choices... "Father Goose"... a whimsical war film, and "Arsenic and Old Lace"... bit of a bias on the latter because I played Cary's role on stage during college.