There basically two types of friction, static friction and dynamic friction.
For Limited Slip applications in differentials, the specialized friction modifier allows for the tuning of dynamic friction in which you have two rotating surfaces needing the correct "grab" or "breakaway" at a certain pressure, torque and rpm.
Shudder or "moaning and groaning" in a differential is when you have stick slip, a condition in which you have partial clutch engagement but the surfaces are still slipping somewhat. This situation generates a lot of heat and wear.
This only applies to cone or clutch LS differentials and the exception is the Torsen or the constant-engagement differential which uses torque biasing of gears.