Both the conjunctions "um" and "damit" introduce a final subordinate sentence. The only difference between the two lies in the subject of the sentence. When using the conjunction "um..zu"  the subject of the subordinate clause must be the same subject of the main clause, whereas when using "damit" the main and the subordinate clause can have two different subjects. 

For example: Ich höre Musik nicht, damit meine Mutter schlafen kann.

The subject of the first clause is "ich", whereas the subject of the subordinate clause is "meine Mutter". For this reason I have used the conjunction "damit" instead of "um...zu".