There really isn't much difference. The only problem that I have is that not all games for the Xbox can be played on the 360. I have one such game, Castlevania: Curse of Darkness.
Basically the 360 uses virtualization software to mimic the original Xbox system so you can play original games. It doesn't always work perfectly though. For example, when I play Forza on my 360, some of the car icons just show up as blurred boxes in the menu's, but are just fine when you are actually racing. Whenever they update the software on the 360, they also add more playable games for the original Xbox, so they are constantly working on it.
I still go back and play Halo, Halo 2, Star Wars Battlefront, and Star Wars Battlefront 2 from time to time.