We use the AF button all the time when focusing the D800 on a tripod because it's a lot more convenient that reaching over and fumbling for the shutter button and then have the chance you push it too far.
Also, in photo mode, when the camera is in continuous focusing mode, holding the AF button with your thumb keeps the focus system active so that you can push the shutter when ever you are ready and the camera is already focused.
AF-ON allows the shooter to Isolate AF from the shutter trigger. For me, its better since it gives me more control over my camera's automatic functions by assigning AF and AE to two separate buttons. I keep my AF set to continuous and my shutter half press activates AE lock. Hold button is set to spot metering, so I can take a spot meter reading if needed, lock in the speed with a half press of the shutter, autofocus with AF-ON, compose, shoot - this is all done in A mode with matrix metering.