A conventional diff uses a cluster of gears, and no clutches to apply power to one wheel only. The wheel with the least traction spins. When going in a straight line forward, it is usually the right wheel. In reverse, it's the left wheel. When in a turn, it's the inside wheel.
The only time really both of them spin is on a snowy surface - you can really see it when doing donuts in a parking lot (my favorite Winter sport!) You can get the back end to really kick out, which only happens if both wheels are spinning.
The traction assistance (control) just applies the brakes to the spinning wheel, causing the other one to spin instead, hopefully helping you to inch along rather being stationary.
Brenda's Crown Vic PI has both trak-lok and traction control, which is pretty rare.
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94 Supra TT Auto - street/strip car
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95 T-Bird LX 4.6 - fully optioned, owned 15 years, 220k miles
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88 T-Bird 3.8 - first T-Bird, owned 5 years, 206k miles