A torsen diff (torque sensing) will unload power to the wheels with the most traction. For example if your front wheels lift of the floor, more power will be sent to the back wheels. This is opposed to the standard type diff where more power would be sent to the front.
Torsen diffs cannot really be tuned like normal diffs.
A spider diff is a normal style diff with more (and smaller) planetary gears inside. This makes them more robust (though the standard ones are very tough) and gives less rotating mass on the drive train.