What happens when a vault dweller levels?

According to the in-game guide their health is based upon their level. With out actual numbers and since the dangers seem to scale along with your dwellers though, this is a bit hard to qualify from an end-user standpoint.. but that is what the guide says leveling does.

Research! I had a room with many different levels and I rushed it until I failed to get the following screen shot

This shows the same damage being done to different level players and there is a clear difference over the length of the encounter. So level up those characters! It keeps them alive longer!

With each level up they gain HP based on their endurance. That means a lvl 50 dweller gaining endurance will not get additional HP. this is likely a bug, but very close to the original Fallout series games.

Test details are listed here: Endurance affects HP only during leveling not (on Reddit)

TL;DR: If you want to max HP for dwellers, train a dweller to Endurance 10 at level 1 before they level, give them endurance buff gear, then level them to 50. Training endurance after they've already levelled won't give them HP, nor will giving outfits that buff endurance to your guards give them more hp after they've already levelled.

Whenever a dweller levels up, they will give you caps, the amount being the same as their level. This means that over time, you will get more and more caps from levelling up your dwellers.

I've not seen any other impact this has. I have one dweller who is level 40 (got him from a breadbox) and he's not noticably different (other than that he arrived with killer stats).