In fact, that's precisely how electronic traction control devices operate.

They use the antilock braking system to brake the wheel that's spinning, inducing power to the non-spinning wheel.

There's actually 5 basic types of differentials in use today:

1) Normal or 'open' differentials: as we've been discussing, and by far the most prevalent.

2) Limited slip: As AnotherScott said, these differentials use clutch packs on the side gears of the differential. Positraction is trade name for this type from Eaton (IIRC, may be Dana..) As torque from the driveshaft increases, outward pressure on the clutches causes them to press against the side gears. This essentially joins the side gears to the differential case causing nearly equal power to be delivered to both driven wheels. The problem with this type, is the locking action is dependent on increasing torque from the driveshaft. Useful in race and high performance car applications, where you want full-throttle traction from the start, with minimal wheelspin.

3) Automatic Lockers: This type uses mechanical devices to lock the side gears to one another based on wheelspin. When one wheel starts to spin, a mechanical lock is engaged sending equal power to both wheels. Detroit Locker is probably to most famous of this type. Useful in 4 wheel drive and off road applications where you don't want to give a lot power from the engine but still need the added traction. The problem is, these are noisy (there's usually a loud CLUNK when the locker engages) and they tend to lock while going around corners (which is extremely dangerous on icy roads) because the outside wheel is travelling faster than the inside wheel (duh).

4) Actuated Lockers: These are similar to the Automatic Locker in that there's a mechanical device which locks the side gears together. However, in this type it is controlled by the driver. The unit functions like an open diff unless actuated by a dashmounted switch. The most famous one of these is from the Austrailian comany ARB and is called the AirLocker. As the name implies, it uses air pressure to lock the gears together. There are now several more variations out there: Toyota has one on newer 4wd trucks and SUVs that uses an electric servo motor; Eaton is coming out with one that uses an electomanetic disc; and a company called OX makes one that uses a shift fork and dash or floor mounted shifter. Cons for this type is cost. An ARB AirLocker carrier will run you $630ish, compressor about $120, and there are, obviously, mods needed to the differential housing itself to plumb the air lines.

5) Spool: There's basically no differential in a spool. The pinion gear drives the ring gear which is mounted on the spool which in turn drives both axles. Good only for drag racing (no turns) really.

More info:
[link|http://www.traction.eaton.com/|Eaton's traction products], in fact I found a better summary than what I've written [link|http://www.traction.eaton.com/product.htm|here]
[link|http://emporium.markwilliams.com/catlist.asp?catid=1004|Mark Williams Spools]
[link|http://www.arb.com.au/|ARB homepage]
[link|http://www.dana.com/tractiontechnologies/|Dana Traction Technologies]
[link|http://www.tractech.com/Products.htm|Tractech (makers of Detroit Locker)]