I'm going to break up my answer into multiple pieces because neither console deserves my judgement of which one is better:
Aesthetics: The first thing that a console has to do to earn my attention is be visually pleasing. The Xbox was slick and cool using brightness to contrast from its Sony counterpart. It was a huge upgrade from the PS3 at the time, which was just an obese mound of black (and the so-called 'slim' version should be called a chubby version). On the 360 it was easy to see the memory card slots, the interchangeable hard-drive and the USB ports, where on the PS3 they were hidden away. In my opinion, Microsoft wins here.
Library: The Xbox didn't have the advantage that Sony did with a library of exclusive titles. Sure they had Halo and Half Life, but these were the only titles that kept them standing in the player dubbed 'console war'. Sony had loads of franchises and to compete had many alternatives to the 360's library. Unchartered was Naughty Dog's attempt at Tomb Raider; Killzone and Resistance were their answers to Gears of War and even had a crack at platform games such as Little Big Planet. Its store wasn't nearly as large as the Xbox 360's, which was packed with arcade classics like Contra and new Indie games like Marble Blast Ultra. I deem this a tie.
Features: The PS3 boasted blu-ray capabilities and DVD playback, whereas Xbox was stuck with the original disc media (can't blame them; I don't see the difference). They also had music playback too, but only the 360 could use this whilst playing games. There was online features and this would be a tie if the PSN didn't get hacked so much and account details were sold to the Soviets. And it wouldn't be until later when the 360 would allow installation of games to make them run faster. It seems this is another tie.
Controllers: My favourite argument of them all, and the final one from me. The Xbox 360 controller is ergodynamic allowing players to pick it up and immediately get to grips with it, something the original didn't allow. But the Dual Shock 3 was a flawed model with slippery triggers, thumbsticks where thumb tendons weren't supposed to go that high and no concaved insides like the original Analogue Controller of the PS1. The battery life on PS3 consoles was better, but wireless was not an option which is a pet peeve of mine. Hands down, the Xbox 360 controller wins my vote.
And that's it for me, because I've run out of characters! ^^