ASC+T is All Season (traction) Control.
I have also heard this called "Automatic Stability Control, plus Traction Control". It basically prevents your car from skidding due to excessive acceleration. It will apply the brakes and reduce engine torque in order to reign in control.
DSC is Dynamic Stability Control. It incorporates the function above, as well as utilizes yaw sensors, wheel speed sensors, and steering angle input, to determine if the car is skidding out of control. This system, one up from ASC+T prevents oversteer, understeer, and just about any other skid you could get yourself into. It can apply any one or more of the brakes, as well as route more engine torque to one front wheel over another.
There is a LOT of info on this... Search
DSC, right here on MINI2.

Chilli Red/White Cooper S, Sport Pack & Sunroof. Delivered June 12th, 2002!
Now with MCS Winter Tire Package.