This is one aspect of iOS that I don't understand or agree with and which seems to be inconsistent between video and sound.
AirPlay only outputs video from apps that specifically provide for "video streaming". What you see on the external screen is only the video stream, not everything that is on the iPad screen.
There is no inbuilt way, in iOS, to output whatever is on the iPad screen to an external monitor as you would find on any other OS. In iOS this is called "video mirroring" and is only available on the iPad 2 after purchasing the AV HDMI adaptor.
When it comes to sound, the situation is different. If you set the iPod app to send sound to AirPlay, all sound, including tap (i.e. OS) sounds etc, are output to the external device even after you exit the iPod app. I find this quite inconsistent and actually annoying.
What I would really like to see "video mirroring" via AirPlay to ATV on all iPad models.
BTW the latest ATV 2 firmware upgrade fixes a small bug. While it was possible to direct sound from the iPod app via the optical (Toslink) output without requiring your TV to be turned on, the TV had to be turned on to initiate this feature. This now seems to have been fixed.