The SPORT button is only for the suspension if you have adaptive drive. (and you only have the SPORT button when u have the adaptive drive option). It does nothing to the engine or gearbox.
Pushing the gear lever to the LEFT puts it in sports mode - and THIS is for the gearbox - drops the gear 1 or 2 down, and keeps it at the best possible gear.
It may also sharpen throttle response and sharper steering (if you have adaptive steering), but i am not sure of this.