No dictionary has ever been able to define difference between 'Complete' and 'Finish'
Here is very good explanation i read online.
When you marry the right woman, you are 'Complete.' If you marry the wrong woman, you are 'Finished.' And , when the right woman catches you with the wrong woman, you are 'Completely Finished.'
Alex above is right. I'll add that "finished" is slang for in danger or in a bad situation/position. The man above is "finished" because marrying the wrong woman will make him miserable.
Another difference between complete and finish is that complete always needs an object after it. This means you have to say WHAT you completed, like a test, task, lesson, etc. You can also use finish to say you finished something, but you can also just say "I finished" when the context is clear (it's obvious what you finished). You cannot just say "I completed," you would have to say "I completed it."
Hope this helps!