They share some qualities, but as the name suggests, Ableton Live is really geared towards live performance use. Logic is more for straight-up production. I mean, people use Ableton Live for production, but I think Logic is more flexible overall. (Because it's simpler, though, Live may be friendlier for people just getting into digital audio production.)
Also, just check out the interfaces: Logic is more for precision work, Ableton is all about getting to your actions quickly, as would be super-necessary when performing live and every millisecond counts.