LIVE is definitely superior to the PSNetwork in a few areas. And I'm not speaking as a fanboy as I own and love both consoles.
One thing that I've noticed is that LIVE is considerably more streamlined than PSN. Whereas downloading stuff from PSN requires multiple screens, followed by the installation to the harddrive, these things are done with fewer button presses and less steps with LIVE. Also, there is no need to request a download be done in the background - they're all done in the background on the Xbox. Again, not a big thing, but it does speed things up considerably. The same applies to system updates.
Sending and receiving messages from friends is also somewhat simpler, as is hooking up for voice or video chat. There's also stuff like 1 vs. 100, which is a fun way to kill some time playing trivia online against everyone in North America (or whatever continent you're situated in). The nice thing about 1 vs. 100 is that it's free to play, and you can win free game downloads and M$ points to spend in the Marketplace.
I will say that I prefer PSN's store setup better than LIVE's Marketplace. With LIVE, video and game downloads are in separate sections, and they only list the 20 or so most recent additions when you select "new releases". But when music games like Rock Band and Guitar Hero are getting dozens of DLC add-on songs per week, a lot of new stuff gets passed over as they simply don't show up in the listings; you have to know about them and do a search to find them.
Finally, due to Microsoft using dedicated servers, many online games run smoother when played on the Xbox versus the PS3. That may all change if and when Sony starts charging for premium content, which could potentially mean dedicated servers of their own. Games like Borderlands run considerably quicker and smoother on the Xbox, for example.