I have also been wondering what goes on in 'sport' mode, apart from the obvious change in shift points and improved throttle response.
I have always had this very distinct 'seat-of -the-pants' feeling, that the engine is actually producing noticeable more power in sport mode. And not just because of the higher rpm, but more power at a given amount of rpm, than you get at the same amount of rpm in 'normal' mode.

Bimmian.com is selling a Sprint Booster Throttle Remapper. And in their description of it, i stumbled over the following:

3) Linearly maps the pedal depression to throttle opening. Press the pedal down 20% and the throttle will open 20%. Press the pedal down 50% and the throttle will open 50%. Without sprint booster, when you depress the pedal 20% the throttle may open 10%, when you press down 40% it may open 25% and when you press down 100% it may open 75%. This is not linear and is not a natural way of driving.

The repeated use of the word "may", indicates that this could be more guessing than hard documented facts. But could they be on to something here?
Does anybody know how much our throttle opens when we mash the pedal into the carpet? -And does it by any chance open more when in 'sport' than in 'normal'? Because that's actually how it feels.

The reason they use may is that it could be a percentage other than they listed but it will it will be some percentage, depends on the vehicle. The stock map is not linear. The reason it is not is to improve economy and make it easier to make the speed changes without heads bouncing off the headrests.

The engine does not produce more HP in sport mode. Assuming those that think the go pedal response is faster, more linear, are correct that will feel like there is more HP.

My sense is Sprint Booster is the most popular but there are several other options that are at other price points because of the list of features are longer or shorter.

At least in that thread posters focused on how it changes the shift program, and didn't mistakenly attribute suspension/steering/throttle pedal response benefits to the transmission sport shifting mode.

I have no numbers of how much more...but the documentation does indicate that there's a difference between the "adaptive" sport mode vs the manually selected sport mode.

BMW only uses simple adjectives "higher" vs "maximum" to distinguish the difference between the tranny's shift pattern in the adaptive sport mode (when in "D") versus the (manually selected) extreme sport mode.

Also note that the documentation about ATC (adaptive transmission control) mentions NOTHING about suspension or steering. This generation of BMWs (e38/e39/e46/e53) doesn't have the same BUS system like the e60/e70/etc. And very few BMW models of this era had a "SPORT" button. The only one that comes to mind is the E39 M5. I'm not sure if the e46 M3 had this feature or not.

ATC (adaptive transmission control) will automatically select 1 of 3 transmission shift programs based on driver influence (dynamics of accelerator pedal movement) upon EACH start-up...and this information/data is NOT stored in the TCM (tranny control module)...thus we get the "adaptive" nature of ATC:

1. extreme economy (granny-toe) mode
2. economy mode
3. sport mode (adaptive sport mode when transmission gear shift is in "D"...but driver is mashing the gas pedal fast and far)