What's the difference between "um...zu" and "damit"?
Both express purpose ("in order to," "so that").
um...zu + Infinitiv: Used when the subject of the main clause and the infinitive clause is the same.
damit + subordinate clause: Used when the subject of the main clause and the subordinate clause is different, or when um...zu is not possible (e.g., with modal verbs in the subordinate clause).
Example
Ich lerne Deutsch, um in Deutschland zu studieren. (Same subject: I study, I study in Germany)
Ich lerne Deutsch, damit meine Freunde mich verstehen. (Different subjects: I study, my friends understand)