It sounds like maybe you're describing AirPlay mirroring instead of passing the video stream straight to the Apple TV via AirPlay from an iPad or iPhone. Mirror

It sounds like maybe you're describing AirPlay mirroring instead of passing the video stream straight to the Apple TV via AirPlay from an iPad or iPhone. Mirroring has to re-encode the on-screen video on the fly, so I've seen the stuttering you mean when mirroring, but I've never experienced that using the built-in video player method to send a video straight to the Apple TV from within the Amazon instant video app, or the Netflix app for that matter.