Traction control only affects your accelleration. It cuts power to the drive wheels to prevent further loss in traction. In the 4th-gen cars, this was cable-activated, and pushed the gas pedal off the floor. In the 5th gen, it'll just cut power. Go to a parking lot in the rain and try to do a burn out and you'll see how it works.
StabiliTrak affects how the car handles. It incorporates the Traction Contral and ABS. If the car is being "pushed" into a turn where a loss of control is detected (or could occur), the "nanny" (system) will step in to employ various steps to keep the vehicle on "trak." This could be applied to brake a front-left wheel while also cutting power to a right-rear wheel, etc. Pretty much if you'd attempt to do some "drifting" or power-slides (fishtail), the StabiliTrak would kick in and keep the car on the "straight and narrow."
Hope this helps.