What is the difference between 출발하다 and 떠나다?
Both can be translated like: Depart.
When can I use one or another?

Question

Answer

A:  Mr.Kim depated Seoul last night.  (move to somewhere)

김선생은 어제 밤 서울을 떠났다.(correct)
                                         출발했다(correct):Probably A is interested in where or when he                      
                                                                           depart from.
 
Twenty minutes from now we will depart. ( I am interested in when it depart.)
지금부터 20분 후에 떠날 것이다. (correctt)
                                 출발할 것이다.(correct)

She waited untill the last of the guests had departed. 
           ( She didn't mind when or where the last gust went to. She  just want  he go out from there.)
그녀는 마지막 손님이 떠날 때까지 기다렸다. (correct)
                                 출발할때까지                (incorrect)

He departed his job December 16. 
그는 12월 16일에 직장을 떠났다.(correct) 
                              출발했다.(incorrect)

He departed this life last year. 
그는 작년에 이 세상을 떠났다. (correct)
                                      출발했다.(incorrect)

Answer

You can distinguish very simply.
출발하다 is depart.
떠나다 is leave.