Limited slip differentials have clutch packs which are engaged by a
difference in torque across the diff. In effect they provide a limited
amount of difference to allow turning etc.
In practice I have found LSD to be crude and next to useless. In cars they
have now been largely superseded by other forms of diff control  or diff
design or electronic traction control acting on the braking system.

Huw

When a car goes around a bend the outside wheel travels further than
the inside. So that drive may be maintained to both wheels a
differential gear is put between them. This puts exactly the same
amount of torque (sort of rotational effort) into each wheel whilst
allowing one to travel further than the other.

This is great when there is a lot of grip but if one wheel loses grip
the maximum torque that can be applied is that which goes to the wheel
which is slipping or spinning. So there are various devices for
maintaining torque to the wheel that still has some grip. The
simplest, locking differentials, have mechanical (dog) clutches that
lock both output shafts together. Plainly once locked the differential
effect is gone and both wheels rotate the same amount on a bend, so
they make driving on firm ground again difficult, not to mention they
accelerate tyre wear.

So a number of more automatic systems are offered. In the racing world
where the spinning is caused by huge amounts of horsepower as the cars
accelerate out of corners it is best to maintain some differential
effect, so as not to foul up the steering, they tend to use limited
slip versions, I am not sure of current types but in the 60s these
would be sprung loaded clutch packs which were forced together by a
ball ann ramp affair which was moved by the relative movement between
the two output shafts.

For your off road driving the lockable type is more useful. A drawback
of locking diffs is that all the drive effort then gets concentrated
on one output shaft, which means these often need beefing up with a
locking diff.

There are other types and ways of doing this, one is by having a
viscous coupling (this was commonly used on the differential between
front and back axles on range rovers), another is by using back to
back one way clutches (no spin diffs) and another is a traction
control that works like ABS brakes. As soon as one wheel starts
spinning a sensor on the wheel notices it is rotating faster than the
others and momentarily applies the brake to that wheel only.

Well from my limited experience for off road use I would say limited
slip is less use than locking which is slightly less good at winding
you out of a hole than "no spin diffs". No spin diffs and articulated
steering work well together but not seen outside construction plant.

IME once you need to lock the diffs you don't get much further than
when you started getting into trouble.

AJH

Different.

LSD locks the diff once one wheel has slipped more than seemed
reasonable to the designers. Locking diffs are controleld by the driver,
they don't lock until you command them to lock.

I have both on my 4x4 (Ford Exploder), it has a choice of "Auto" which
is normal driving where the drive is 100% to the rears. The rear diff is
a LSD so if one wheel slips the other will take up the drive sooner or
later. The centre diff also has limited slip as does the front diff.
Eventually if enough slipp happens all the diffs lockup and all wheels
are drive at the same speed.

Sounds good, but in practice it's a PITA as when it does cut in, it cuts
in with a bang and is likely to cause a four wheel skid. The better
option is to dial the required setting in using the dashboard selector,
as 4x4 High ot 4x4 low which works pretty much like the Land Rover
(Series III) lever. Drive to all wheels, all diffs locked.