As stated above PS2 is last generation technology. And there is a huge difference in graphics, processing speed not to mention the PS3 could also double as your families Blu-Ray player if you do not already have one. It also comes in handy if you like to stream media from your PC to the TV it is hooked up to. And the built in wireless makes it more mobile than a PS2 which has to be where the internet connection is (assuming your house has wireless of course).
But to take full advantage I would say you should at least have a TV capable of 720P (preferably 1080i or 1080P but at least some form of hi-def TV). Otherwise the games will be better but you will not get much benefit of better graphics.