"He helped me" is relatively neutral and simply means "He gave me assistance". "He helped me out" is stronger and implies "He helped me and - it made a real difference - I really appreciated it - I was able to fix the problem.

"He helped me" can be used as part of a phrase, like "he helped me do x." "He helped me move," for example. "He helped me out" is typically a phrase that stands alone. However, we can say "he helped me out *with* x." "He helped me out with my move."

he helped me out

Everyone's answers are correct, but they are leaving out the context of why you would use "out" at all. "He helped me out" is a shortened idiom for: He helped me get out of a bad situation. I was stuck in a well. He helped me (get) out (of the well). My car had a flat. He helped me (get) out (of that situation). (By fixing the flat.) My wife was in labor and our dog got loose. He helped us out (by catching the dog). And so on. It is stronger than "he helped me" because it implies that person or persons were solely responsible for fixing your problem.