I think you mean Xbox 360 rather than just Xbox.
Xbox came out in 2001, 5 years before PS3 was launched so of course PS3 has better graphics.
Now in general you can refer to either an Xbox Original or Xbox 360 as "an Xbox" just like you can call a Toyota Prius and Toyota Prado "a Toyota". Same with PS1, PS2 and PS3 referred to any one as as "a Playstation"
But when you say PS3 and Xbox together, it seems like you are specifically referring to the Xbox 1 rather than Xbox 360.
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Actually, if the full extent of your console gaming experience is on the PS2, and you don't have a HDTV nor plan to buy one, I suggest that you get yourself an Xbox Original (that's the First Xbox) rather than a PS3 or even an Xbox-360 if only because Xbox 1 still has SIGNIFICANTLY better graphics than PS2 and an amazing catalogue of games.
It cheap as hell, you can get an Xbox console for as little as $20 and games for as little as $5 since it is an "obsolete" console. But Xbox Live still works for the Xbox original, you can still play online and get updates. Here are some games you should check out that you just can't get on PS2:
-Ninja Gaiden Black
-Forza motorsport
-Chronicles of Riddick
-Crimson Skies
-Halo: Combat Evolved (co-op/multiplayer)
-Halo 2 (co-op/multiplayer)
-Fable: The Lost Chapters
-Far Cry Instincts
-House of the Dead 2 & 3
-Half Life 2
-Counterstrike
-Doom 3 <-----you won't believe how good this one looks
-Star Wars: Republic Commando
-Star Wars: Knights of the Old Republic 1 & 2
-Star Wars Jedi Academy / Jedi Outcast
-Unreal Championship 1 & 2
-Rainbow Six
-Hell Wars (Painkiller as reviewed by yahtzee but renamed for dumb copyright reasons)
-MechAssault 1 & 2
Some of those are on PC too, but if you can't be bothered with PC, this is a good alternative if you want to avoid trying to figure out installing games. Just put in the disk and play.
Also these games are also on PS2 but look SIGNIFICANTLY better on Xbox often with many extra features:
-Splinter Cell: Chaos Theory (co-op/multiplayer)
-GTA: Double Pack
-GTA: San Andreas
-Burnout 3: Takedown
-Return To Castle Wolfenstein (co-op and online, lacking on PS2)
-Psychonauts
-Black
-Mercenaries: Playground of destruction
There is NO POINT in moving onto the next generation if you haven't used the full potential of the last generation. Remember, Xbox console, games and accessories are dirt cheap but it will still all be A LOT better than anything you've had on PS2 so far, I guarantee it.
I strongly suggest you try the component output cables if your TV can take them. It makes such an amazing difference.
Also, PS3 is not only still quite expensive at $300 but a HDTV you'll need to fully appreciate the games = $500 at least. I would not even recommend the Xbox 360 yet unless you really have the money for it, gaming at the moment is expensive at $60 per game + DLC.