What is the difference bewteen "um zu" and "damit"?

Both "um zu" and "damit" are used to express a goal, a purpose.

The question asked before often starts with "Wofür", "Wozu?", "Mit welchem Ziel?". 

"Um zu" is an infinitive construction: for example Ich fahre nach Berlin, um mein Familie zu sehen. Um comes after the comma and zu before the verb. When the subject remains the same in both parts of the sentence, we use "um zu" rather than "damit". Whenever you can replace the English "to" with "in order to" indicates that you have to use "um zu". 

"Damit" on the other hand is a subordinate clause. For example, Ich studiere viel, damit meine Mutter stolz auf mich ist. "Ich" and "Mein Mutter" are different subjects. "Damit" in English means "so that".