Both are great systems, with their own pros and cons in both hardware and games.
360 is cheaper with much larger games library and 3rd party support from games companies/ peripheral manufacturers but u have to pay for Xbox Live to play games online (although you do get 1 month free when you make a new account). Although it is cheaper for a base system you will have to pay for EVERYTHING else you need - £32 for a joypad, £15 for the joypad battery pack and charging lead, £50 for the wireless adaptor...don't be tricked by the ultra cheap starting price, costs will soon start to add when you start totaling up all the added extras you might want. Also the 360 has very little homebrew applictations yet, so no emulators like on a modded old xbox. Thats the result of MS being able to sell old arcade games etc for hard earned cash on xbox live unfortunately.
PS3 has Blu Ray, an updated version of the classic Playstation joypad that everybody feels comfortable using, with the added support of motion sensing built into the pads - i LOVE PS3 pads, they just work....perfectly!!!! Built in wireless, free online gameplay (AFAIK) and potential for a lot of 3rd party applications to come forward if the developers choose to pursue it. The extra hardware capabilities reflect the higher price tag off the shelf, it just depends what you want your console for....
Both systems have games that will NOT appear on the other console ie: Gran Turismo for PS3, Halo / Gears of War for 360. This could very well be the deal sealer for many a person looking at buying one of these consoles.
Yes it is true that the older 360's especially had a problem with the 'Red Ring Of Death' (RRoD) although this has given me the opportunity to pick up extremely cheap broken 360s and give them an easy repair :-) On the flip side of this (sorry to all the Sony fanboys out there) the PS3 has also had isolated issues of a problem called the 'Yellow Light Of Death' (YLoD) although not anywhere on the scale as the 360. Seems Sony make slip ups too! Ive not looked into the fix for the YLoD but I do know one is out there on the internet - Sony want £150 to repair it themselves!!!!! This is quite a shock considering the cost of fixing the 360's RRoD yourself which adds up to a total of a few pence...
There are very minor differences in the way that both consoles handle the graphics in games, it is the Anti Aliasing, Texture Filtering and Vertical Sync options that vary most but to the normal eye these differences will largely be completely unnoticable unless you are actually looking for them and if you don't know what these graphics functions are then you almost definately will not notice the difference!
There are also differences in the consoles native display resolution (720/1080 i/p) but id be chatting about something im not 100% sure on if I went into that.
Those are just a few things I can think of off the top of my head, im sure other people will be able to elaborate on what ive said. My own personal opinion is as follows:
If you dont need Blu Ray and dont need a console for the extremely stylish way it looks then go with a 360. A 360 will not burn as big a hole in your pocket and can play games just as good, if not better than the PS3. Xbox Live (despite the need to be payed for) is a solid platform for online gaming and has proved to be a winner in the marketplace.
If you want a Blu Ray player and are looking for a console as well then there really is no choice, the PS3 wins hands down in this case...just make sure you have a 1080p screen ready to utilise it beforehand!
I am by no means an expert but i hope this helps a bit.
*edit: I notice a lot of people quoting the PS3 has better graphics - sacrificing anti aliasing and texture filtering for vertical sync does not equate to better graphics... It really is swings and roundabouts depending on the game at hand. Coming from a PC gaming background i can see that, neither consoles are mind numbingly brilliant at graphics with Call Of Duty 4 looking better and running faster on a 4-5 year old PC than on either console. Games like GTA4, Rainbow Six Vegas, Splinter Cell Double Agent, Assassins Creed etc only run better on the consoles due to poor programming on the PC ports of the game (funny enough, 3 of those titles mentioned were developed by Ubisoft...coincidence?) This is besides the point though, just thought id add the info for anybody that cares to read :-)