Yeah, vudu's right, the GameBoy Player is the *official* product, while the Game Port isn't. While I don't have the Game Port (I'm waiting to get one *cheap*, just for 'collecting' reasons), I can say that I'm mucho happy with my GameBoy Player.
I should also note that the GameBoy Player features a link port that allows you to link to other GameBoys/GameBoy Colors/Game Boy Advance/SP/other GameBoy Players - or even other GameCubes (for connectivity purposes). Also, with the GameBoy Player, if you want, you can use a GCN Controller or, with a GBA/GCN cable, you can use a GBA/SP as a controller. The Advance Game Port does not allow any of this.