They each have their own ups and downs but in the end none of them will necessarily make your music sound any better.
I don't know much about Logic, all I know is that it's relatively cheap and I think it's only for Mac.
I've used FL Studio for nearly four years however, so naturally I recommend it. It's not too expensive (you can even get it here for a 10% discount), has what is, in my opinion, a brilliant and user-friendly interface, a very impressive piano roll and most people use it so it's always incredibly easy to find tutorials, people who might be able to help you if you have any issues or people to collaborate with. Best of all, if you buy it you've got lifetime free updates. That means when they launch FL Studio 11, 12, etc... they'll be all for free. Do note that FL is only compatible with Windows though.
However, if you're more into live playing/recording, then Ableton Live is your program. As its name implies, it's great for live stuff and while it doesn't have lifetime free updates, I haven't heard anything bad about it. I know my Physics teacher uses it and I'm sure he'd recommend it.
But do remember that like the above people said, you can't call one program better than the other with these high-end DAWs. And even if you can, they will not improve your music. They might make certain things easier than others, but they will not improve your music since you are the only one who can do that.