This thread has the potential of becoming a massive flame war, but I'll give as fair and helpful an answer as I can. Just keep in mind that I don't actually own an Xbox 360 but do own a PS3.
The PS3, in my opinion, has a higher quality, more diverse and larger library of exclusives.
There's the
Uncharted series,
Metal Gear Solid 4 and (arguably)
God of War 3 for AAA games.
There's the
Resistance and
Killzone series for solid First Person Shooters
There's the
Infamous and
Ratchet & Clank series for really good Third Person Shooters and Platformers.
Heavy Rain and the
LittleBigPlanet series are more innovative than any current Xbox 360 exclusives (again, in my opinion).
Other notable PS3 exclusive games are the
Motorstorm series,
Valkyria Chronicles,
Demon's Souls and
Warhawk. There are also new editions to the
Uncharted,
Warhawk and
Resistance series coming out in the near future, as well as
Last Guardian, the next Team Ico game (although not much as been said about that game for quite a while).
There are probably still a few notable exclusive games that I have failed to mention.
Overall, the PS3 has a very good exclusive library of games.
In terms of online capability, the PS3 has the advantage of having a free online service, although it isn't quite as robust as the XBOX 360's online. The Playstation store probably isn't quite as good as the Xbox Live Arcade (although I'm not sure as I don't have a 360), but the PS3 still has some very good exclusive downloadable games, such as
Fl0w and
Flower, the
Pixeljunk series,
PAIN,
Dead Nation,
Echochrome,
Super Stardust HD and
Everyday Shooter (which is also available on PC). You're less likely to encounter annoying immature gamers while playing online on PSN, but most people also don't have mics, so you're pretty much trading being called a ****** all the time with mostly silence. Also, since less people have PS3s than 360s, you'll probably have less people in your circle of friends to play online with.
The PS3 can also play Blu-Rays, obviously. Playstation move is also there if you're interested in more casual gaming. Although the PS3 does have a web browser, it's very slow and fairly redundant as long as you have a half-decent PC or laptop. The PS3 will probably end up having better 3D support if you're interested in that.
Overall, I'd recommend getting a PS3, but I have to admit to being biased towards the PS3 originally.
On a side note, Google Chrome's spell checker really hates PS3 game titles. That might be a point in the 360s favour
