I'd like to add a bit about the difference between finish and complete, in hope that it can help "complete" other answers, which are already good, a little. (In other words, I write this as a supplementary answer.)
The striking difference between the two can be observed in the contrast of "You finish me" and "You complete me".
"You finish me" (or "You finish me off") means "You kill me", i.e. you put "the end" to my life.
"You complete me" is normally used between couples, in the sense that one (e.g. a wife) completes the other (e.g. a husband). With her, he is a more "complete" person. Without her, he is "incomplete". A good example of this phrase can be found in Jerry Maguire (one of my favorite movies of all time!):
Jerry Maguire: [babbling and struggling] I love you. You... you complete me. And I just...
Dorothy: Shut up,
Dorothy: just shut up.
Dorothy: You had me at "hello". You had me at "hello".
Source: IMDb http://www.imdb.com/title/tt0116695/quotes?item=qt0389299
So, once you finish me, my life is "no more".
But once you complete me, my life is "perfect".
Back to your homework sentence (He completed/finished his homework), homework is a kind of work. After work reaches its perfect state (i.e. "it's done"), we'll have no more of that work. This is why in the context of work, including your homework, saying either He finished his homework or He completed his homework will have pretty much the same meaning.