These two photos show the destructive damage
that both oil and no oil can do.  

The above left photo is the result of a Ford 9" differential being filled with the wrong oil. Wrong oil in this case being an oil that has no "extreme pressure" additives to accommodate the surface pressures during contact under load. The gear-set will just wear away till you could nearly shave yourself with the pinion teeth.

In the right-hand photo, the Toyota Hi-Lux pinion is on it's way to becoming a molten blob. The differential has run out of oil, but the driver has not realized it and has kept on driving. With no oil in the differential, there will be no heat dissipation and consequently the differential internal temperature will continue to rise. As the temperature rises, the metal will become softer and parts will begin to weld to each other, thus eliminating component co-operation. Complete seizure will be the result.