It's only a bit of software and only for show.
The f20/21 1 series auto only has a single automatic gearbox the ZF 8HP45. That's if you have a 116i or M135i.
If you pay the extra £110 for sports auto you get steering wheel paddles and changes to the software that prevents up and down changes during manual mode changes so you can redline the engine if you want. You don't get a different gearbox, just paddles and software changes.
If you down shift with the standard software gearbox the revs are increased to match engine speed with the rotational speed of the lower gear. On the sports auto the revs are increased further to give a sporty blip on down changes.