Bad Ring Gear
For comparison, this picture shows a ring gear that might work, after a fashion, but you don't want to use it. (Stock photo; it's not mine.) The gear teeth should be shiny and smooth, but here you see pronounced striations. If this were put into service, it would be noisy and wear the pinion rapidly. The noise would then increase until it either became intolerable or something failed.
The pinion didn't look as bad, but since the ring and pinion gears are produced as sets, it's also useless. Differentials are pretty tough; take care of yours, replace the oil regularly, and this will never happen.