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As you can see from 3318C5's link, in Competitive Driving mode the computer will allow wheel spin (Traction Control is OFF), but will take steps to keep the car in a "pointed where you want to go" direction (Active Handling is turned ON).
It is pretty easy to test this.
Make sure you are somewhere where you cannot harm/injure anything/one.
Have the car in default mode (Active Handling ON-Traction Control ON) and floor it from a dead stop. Any wheel spin will quickly be arrested by the traction Control (brakes may be applied, engine RPMs may be reduced), and any deviation from the straight and narrow direction-wise and Active Handling will make serious attempts by engine and brake adjustments to keep the car on the straight and narrow.
Now, press the little button for 5+ seconds and put the car in Competitive Driving mode, and do the same. (Traction Control OFF, Active Handling ON). You will most likely experience wheels spin, and if you keep you foot to the floor, you will most likely experience Active Handling making adjustments to keep the car from going sideways.
Now, press the little button until the skidding Corvette icon is illuminated on the dash (Active Handling OFF Traction Control OFF), and do the same procedure. You may be surprised to learn how much "work" the Active Handling has been doing.