Actually it doesn't do anything amazing. It just simply cuts the power. Unlike a computer, the XBox doesn't have any major work that needs to be done when switching off. There is no data that needs to be saved, and there is no processes that really need to be quit cleanly.
The XBox was built with a different concept in mind then a PC, and therefore, just pressing the button, pulling the power or even using the control to shutdown the console, all do the same thing, cut the power. All important data on the XBox is saved at the point it needs to be saved, and it hardly if ever, keeps important data in memory.
At the time of writing the answer I forgot about the scenario of active downloads. If you have configured you XBox 360 to continue downloading if when off, the XBox will go into a sleep state when using the controller to switched off, with the exception of pulling the power.