Can a rear differential be repaired?
First time question; long time listener and NPR contributor
Possibly.
What is wrong with the diff?
Yes. Unless the damage is extensive enough that it actually broke the housing that holds the differential.
If the housing is undamaged but the ring and pinion gears are damaged then a used unit from the salvage might be a better option as the costs of ring/pinion gears, bearings, seals, and labor on a fairly technical job could be somewhat high.
Rear differential repairs are very expensive. Some years ago I had the bearings replaced on a Chevy rear end at the cost of $440. No gears were damaged. If your gears are damaged, the better approach would be to find a wreck and get the wholes unit. In my case, a whole rear axle assembly would only have cost $750, clearly the way to go if there is significant damage on the gears. Our family doctor had to fix the rear end of his C class Mercedes and it came to $2800!