To throw in a new point, beside all the discussions about story and whatever: I think there is something that the so-called "next-gen" shooter games like Crysis do better than (most of) the old-school shooter games: They are very intense, you really feel like you're in the game. I know that is mainly because of the graphics, but it's not just the graphics.
For example, F.E.A.R (both the new and the original one) has very good particle effects and nice physics that make even just running through the levels while shooting a funny thing; In a fight then, you're just thinking "wow" because crates and barrels are flying around you, sparks and wall fractures are scattering all over the floor and explosions are blasting everything away.
This can also be found in Crysis, Far Cry 2 and nearly all modern first person shooter games (as well as in most third person games like GTA 4).
Another thing: Try and start one of these old-school FPS, for example Unreal, and take a look down. What do you see? I can tell you, nothing exept for the floor.
Now start, say, F.E.A.R. Now look down. You can see the body of the main character as well as his shadow.
This is as well giving the player the feel of really being in the game, and it's also something you cannot find in most of the older FPS.

Well, but that's just what I think, others may have other opinions.