Exalt means "to raise up," "to esteem greatly," or "to glorify."
Note that it has the same root as altitude. The root alt- means "high."
Exult means "to rejoice" or "to be openly happy about."
There is a sentence in the Bible which, in one translation, uses both words.
Example: And Mary said, "My soul exalts the Lord, and my spirit has exulted in God my Savior." --Luke 1:46,47