Who would win?

In hand to hand combat, Jason Bourne would win easily due to his martial arts training in Krav Maga and Escrima. Let's not forget his Jackie Chan like ability to improv objects that could be used as weapons. Bond seems to rely more on brute strength and less on technique.

Mentally speaking, they're both excellent. However, I think Bourne edge out a little. Remember that scene where he's in a diner with Marie and he tells her about all of his observations? Nicky also stated in the second film, "It's not a mistake. They don't make mistakes. They don't do random. There's always an objective. Always a target." Bond doesn't have this mentallity given his promiscuous nature.

In a straight out gun fight? No clue I guess they would be even.

It's like comparing Tom Cruise to Arnold Scwarznegger, as if anyone could beat Bond.

In previous films Bond has always been immortal, but in the latest Casino Royale he was portrayed as vulnerable for the first time, although he did chase down that parkour guy pretty well.

I would have to say Bond, Bourne has just never really impressed me. He is more hide and seek than anything else.

Bond is a man (at least played by Connery, Dalton and Craig) Matt Damon is still seems like a little boy.

I doubt many people would argue that a 77 year old Connery could crush a 37 year old Damon :D

Bond. No question about it.