well I don't have a dog in this fight, quite frankly mine is sleeping on my slippers actually
( pictured below )
but as far I know LEDguy explained it well, eventhough alpg88 was ( at least I feel like ) saying the same thing at first...?!
So to my current knowledge modern SUVs or four-wheel drive cars have open diffs all around ( mostly ) so when a wheel on the same axle ( e.g. front ) starts spinning due to lack of traction, the TC detects that, applies brake to that wheel, to sort of replicate a locked diff, giving torque to the non-spinning wheel. An improvement to this is the "sperrdifferential" as we call it here, ot auto-locking diffs / limited slip differentials, which work as an open differential most of the time, but when one of the wheels starts spinning then the wheels mechanically lock together ( either visco or torsen ). And the most "advanced" is when you can manually lock diffs together, either pneumatically or hydraulically, allowing you to decide on your own when and what axles you need locked together, this is the most expensive and rare, but there are kits to retrofit open diffs to be lockable ( most I've seen are pneumatic tho ).
So to sum up, nowadays usually the TC will act as a locking or auto-locking diff, with the difference that a TC can/will limit power to the wheels, so it will feel like you're not going as fast as you can, but much safer than a locking diff like in racecars, where you can plow out of the turn full force.
Or at least this is my understanding, and from this I believe having just TC is the most common and safest way manufacturers make sure that the driver will stay on the road.
Ps.: From my own experience a TC on icy/slippery roads feel like you are held back quite substantially, but also gives you a railcar experience, while limited slip diffs give you a more "fun" experience on traction loss, but can surprise you on corner exits ( a'la BMW ).
Oh and to give an input on Poppy's question, I think you would want the least amount of weight on a car for increased stopping performance, but you'd also want to have a good balance of weight so the rear won't snap out on braking.