Gear shifter in sport mode/drive mode

[Henkan]

What is the difference/interplay/connection between the drive modes and the gear mode? Ie when you push the gear shifter to the left? The gear immediately shifts down one gear but that also happens when changing to sports mode. I understand that a lot of other stuff happens in sport mode too but I am interested in the gear behavior primarily. What more happens if you are in sport mode and change the gear to sport too?

[Qsilver7]

The gear shift lever's S/M modes only affect the car's engine and transmission (drivetrain).

The driving experience button's sport mode affects not only the engine & transmission (drivetrain), but also the suspension & steering (chassis). And especially the suspension mapping if you have any of the adaptive suspension options.

The chart below gives some info as to how the engine/tranny/suspension/steering mapping is affected by the various options available to BMW drivers when using the gear shift lever positions and driving experience "mode" buttons: