I own both consoles and a multitude of games for each and, although I prefer my PS3 for many many things, I'm going to have to say Achievements. Gamerscore gives a better feeling of progression and competition with friends, it's a more flexible way of showing how much your effort for that particular accomplishment was worth (a scoring system that can range from 5 to 500 is far more telling that one that just shows gold, silver or bronze) and the Achievement notification sound is positively iconic.
Take into consideration that Trophies are an unabashed rip-off of Achievements in the first place, add to that the fact that everyone, wether they own a 360 or not knows the Achievement tone (if you say you don't, you're lying) and no one who doesn't own a PS3 knows what a Trophy sounds like, and you have a clear victor in the race for best entirely unecessary accomplishment tracking system.