Björk (or actually her crew) has both Live and Logic on stage: http://www.apple.com/pro/profiles/pollard/index2.html
This is a question I'm constantly thinking about. There are really annoying things with Live. But on the other hand, I've used Logic and Cubase before and the ALSO have really annoying things. You can't have it all in one program, some other program almost always does some of your favourite features better.
I think Live is a better composition tool. I never realized this until someone wrote it, but that's really true. ProTools or Logic are probably better for constant recording work and of course you will have to record too as a musician and at those recording moments you'll maybe feel irritated at certain features missing from Live. But then those everyday moments when you're NOT recording in a studio and you DON'T need vocal comping, keyboard shortcuts to activate/deactivate clips etc etc, you'll find out that working with Live is a real joy. It's truely inspirating piece of software and what it does, it does it in a really innovative and easy way. It's really quick and easy to get the sound you want out of Live.
And like with all the software, it gets better every year (guys, don't start with the bugs+beta-conversation....)
So my verdict is (too): for songwriting, producing and especially composing and innovating in music, Live is really cool. But if you work really much with video and need abilities to stretch midi to video, work with timecode etc etc, that's an area where Logic wins hands down. I'm also studying film music and so far I've gotten along with Live, too, but Logic does have some very compelling features here. Also an in-built notation module, which is quite cool... although it seems to be quite a bit harder to use than a specialized notation software.