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For engine - the throttle mapping is more responsive in Sport and even more so in Sport+ mode. Additionally, the car should make more power in Sport/Sport+ mode - you may even notice the oil temperature goes up in Sport/Sport+ mode (this was true on the F30 340i with MPPSK - the extra power was only available in Sport/Sport+ mode - I maybe assuming the same for the M340i but the butt dyno certainly makes it feel that way).
Transmission - this holds the gears longer in Automatic mode and it will downshift the car as you slow down, but it also definitely speeds up and firms up the shifts. There is a marked difference imho between Sport and Sport+ for Transmission mode in terms of how firm the shifts are.
Pushing the shifter to the left puts the car in "Sport Transmission" mode which definitely allows the car to hold gears longer and downshift as you slow down but if you press one of the paddles behind the steering wheel it will also allow the car to stay in Manual mode, meaning you control the shifts and it won't switch back to automatic mode.
If I'm in Sport+ mode I usually also drive in Manual mode.
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