As far as I know, there were some games that were 'optimized' for the Gameboy Player, meaning that if you played these games on the Gameboy Player, you would experience brighter graphics, etc. There was a palette usually assigned just for the Gameboy Player.
If I recall, the games would have a calling card at start-up that said "Gameboy Player". I remember also that some of the games added RUMBLE when you would play them on the Gameboy Player with your Gamecube Controller. It would rumble, which was a kinda neat feature. Mario & Luigi: Superstar Saga was one of these.