I have it on the Xbox and haven't taken it on Live yet.
The graphics are incredible.
The music is different, but the cues could be better and maybe not fade in and out so sudden.
The rag doll effects are a vast improvement.
The new moves and interactions are brilliant.
It feels like an entirely new game and you can see the evolution from the previous two.
There are so many details in this game that add to the atmosphere and immersion and, in my opinion, make it by far the best in the series.
A few I like are when you walk by people Sam moves his head to look at them, when you walk back from an enemy he actually steps backwards rather then turning around and as you get closer to people Sam's pose changes and he reaches for his knife ready for if you grab them (he always holds the knife to their throat when you grab people).
The enemy A.I. is definitely the most intelligent yet, but still has some flaws.
The game gets better and better as the levels go on.