When a dweller levels up, what actually changes? My thought at first was that they gain increased HP, but after reading this answer I noticed that HP seems to be controlled by Endurance.
Well, the game's own wiki says levelling up a dweller actually increases his health points and since there's no way to actually check how many health points a dweller has before and after levelling up, I'm just gonna trust the wiki :)
Also the wiki says the endurance will help a dweller survive more in the wasteland... this can be reflected by him having more health points or by him just being less prone to lose health points (so even though 2 dwellers have the same number of health points, the one with better endurance loses less health points when fight a rat, a scorpion or something like that)
TL/DR: Levelling up increases health points, Endurance makes dweller lose less health points when fights a monster.
A side benefit of leveling up is that each subsequent level gains more caps. Leveling to 30 gives 30 caps, 31 gives 31 caps, etc.