there is much difference between "시작" and "출발". literally, "시작" means, as you said, "start" or" begin", it's 开始 in chinese while "출발" is used to mean "to depart or to leave" and "出发“in mandarin. also they can be translated in lots of ways depending on contexts in English as every language does.

a little different for example) 시작 = game start, let's start. 출발 = go (walk,run,drive, etc) if you wanna go anywhere you can use it