Live on the 360 is leaps and bounds ahead of PS3 Online.
The main issue with the PS3's online service, is it's not a unified structure. Each game is handled seperately by it's respective developer/publisher. The same way it was on the PS2. While the PS3's online is better than what the PS2 offered, Sony still has a HUGE hill to climb if they hope to ever reach the level of Xbox Live.
The biggest issue with the PS3's online service is there isn't any type of real unified system. No PS3 Online functionalities are intergrated into the games....meaning, they each use their own system. You have to log all the way out of a game, to the main menu to see if your friends are on. If you send them an invite, you have to hope they're playing the same game....or else they won't get it. But there's no way for you to know what they're doing....just that they're online. But even if you get an invite...you can't just accept it. You have to actually go search for the game and join...it's more like a note that your friends are playing this certain game. This among other things is a big hurdle they have to overcome.
On the 360, the online, and community features are done very well. All the friends lists are intergrated into one full system on every game, across the board. You can see who is online, and what they're doing at anytime. You can send invites to anyone at anytime, and it will jump them right to your lobby. If they are playing a different game, they'll still get an invite that will allow them to switch the game, and then jump right in with you. New friends you make in games will be transferred to your friends list, there's a ratings system that carries over, there's match making based on your category/skills/zone, Gamerscores, in-game video/voice/text messaging, etc., etc. Xbox Live offers sooooo much more just on the multiplayer game level. Not to mention the other areas of Live.