with the classic controller, it acts as the controller with ALL of the VC games. The games from NES of course don't need the classic controller plugged in to work, because you can turn the remote itself on the side and use it like the good 'ol NES controller. a few other games outside of the NES also work with the remote sideways, but i don't remember which ones, you'll have to look to see what controllers are available.
With the adapters, i think the majority of them are wired, so you have to be careful if you get excited and yank the controller when you play, but with the classic controller, It's wireless except for the connector to the remote. I like that a lot better GC controller trying to play VC games. now i can just lay down. Yes, i am lazy by nature. you COULD get a cable extender for the GC controller, but thats the point if you could go wireless.
As for the controls, i think they're realativly the same. I enjoy the classic controller much more because it feels like the SNES controller, which reminds me of my childhood
Also, you can move the cursor on the screen with the classic controller without having to pick up the remote itself. there is a home button built in, and it just makes it easier. You can pick up the remote of course, but the controller is just as capable. I don't think the GC controller can do that.
Overall, i like the classic controller better, and i seldom use the GC controller except to play GC games of course.