To use the wireless xbox controller you need the adapter. Just because you connect the charge cable to the USB on the computer doesn't mean the controller still isn't wireless.
The xbox controllers are always wireless, their signal is never sent through the charge cable. So the Adapter allows you to sync your wireless controller to the PC and then the game should pick it up as an xbox controller and auto config the buttons for you, NFS: Hot Pursuit did for me.
You could also just get a wired xbox controller from wal-mart or a normal PC controller if you prefer, they generally have the same design.