This is a debate that could get pretty wild, though being on a european-based forum, there's probably some justifiable bias.
Having sat on both sides of the fence, I'll throw in my 2 pence worth.
Katanas have incredible heat treatment properties that would rival nearly all european blades. There's an extremely impressive japanese swordsman, Isao Machii, who has done some pretty insane things with a katana. Including cutting a steel pipe and cutting through a steel sheet without damaging the sword (
). There's lots more on him if you google; the plastic bullet and the pea are jaw-dropping. Swordsman aside, the soft back of the katana is what allows the sharpened, hard edge, to stay that way without cracking. They will cut like there's no tomorrow, but that's about as far as it gets; they're single edged and not designed for thrusting, either.
European swords (longswords in particular) aren't likely to get cut through. Bent could be a mildly possible expectation, but you're talking about varying angles of impact, whether or not the holder is gripping and pushing the sword so hard it doesn't knock back into him before bending, etc. They are also great cutters, and their utility lies in the double edge and the easy-to-stab-with tip. "Combos", as we usually think of them, have extreme variety and practicality with a longsword.
Mythbusters did an episode a while back about a katana that cut through a gun. They never got it to work, and broke a few blades trying, if I remember right. Even with the gun barrel softened, it didn't work - and who would go into battle with a squishy sword?
In the end, a longsword would be awkward in japanese context, and a katana would be awkward in european context. I think the most reasonable thing I can say is that each weapon is designed with a specific intent when battling in a specific region and time period with specific styles of armor and combat - and each weapon excelled at performing it's specific intent.
Either one would do horrible, unspeakable things to an unarmored opponent with a half-decent strike. Dead is dead, as they say.
(I think the video is from a History Channel special called "The Samurai". Been a while since I saw it.)