You don't start the game any different. It does spin the disc to make sure you have the actual game in the drive though. It is much quieter and produces slightly less heat in your XBOX. Not to mention that it reduces wear and tear on your disc drive.
On a side note, I have installed tons of games to my HD. When I no longer use or own a game that I have installed to my HD, I delete them through memory management.
Last year, my Madden 10 disc got so scratched it wouldn't run. I realized that it would still work if I installed it to the HDD, so I did, and it ran perfectly after that. You get faster load times and slightly better FPS.