Not really. It's like putting a playstation2 game inside your Xbox 360. The first Xbox and playstation2 have like the same games for each other and same quality when Xbox 360 is a more advanced game system. but there are a couple of regular Xbox games that you can buy online to play on the 360. like for example, grand theft auto San Andreas is available online for the 360.
It is possible to use your games from the original Xbox on the new Xbox 360. However, not all games from the original Xbox are compatible with the new Xbox 360. You can check some of the games that you can play at http://xbox.about.com/od/xbox360faqs/a/360bclist.htm.