To exalt, means to glorify or elevate something, but to exult is to rejoice. Exalt your favorite pro-wrestler, Jesus, or your status in the world. Exult when you get the last two tickets to see your favorite band.
Exalt is a transitive verb, so it needs a direct object: you have to exalt something. If we glorify God, we exalt him. It can also mean to heighten something, to increase its effects. You can exalt, or raise, the Virgin Mary, or a level, like in these examples:
His devotional writings, which exalt Mary by superstitious tales of miracles, were extremely popular in all Catholic countries. (J.H. Kurtz)
At a less exalted level, we buy and sell attention all the time, usually as part of some other transaction. (Slate)
I'm afraid, though, that very few people walk on that exalted plane. (New York Times)
To exult (with a "u") is to whoop it up, celebrate, to be happy. If you rejoice at your team's big win, you exult in it. It's an intransitive verb, so it doesn't need an object — you exult all by yourself, any way you want:
Little wonder Ferry's fans were out in force Tuesday, but they had little reason to exult as the show opened in subdued style. (Chicago Tribune)
We do have our problems and we worry and whine and complain, but we also have to celebrate and exult. (Associated Press)
Coffers will bulge, fans will exult, civilization will fail to collapse. (New York Times)
If you think something is the best thing ever and you practically worship it, you exalt it. You have to exalt something. If you're just thrilled, and your sentence doesn't require a direct object, choose exult instead.