By default, single-tapping and double-tapping on the Home button do the same thing. However, you can assign different functionality to double-tapping.
You can assign any open paper smartpen app (apps that do not require paper buttons) or Quick Command. For instance, if you do a lot of calculations, you could assign the Calc Quick Command to launch with a double-tap on the Home button.
The smartpen screen prompts you to write an app name or a Quick Command.