I think of "have to" as something that you must do no matter what. You do not have the choice to do it. "I have to go." -> Very strong emphasis on going. You CANNOT stay. Whereas, "should" is more a choice or a suggestion. You need to do something but at the same time you can still choose to do it or not. "I should go."-> I have a choice to make. Even though I have something I need to do, maybe I don't want to go right now. You should do it. You have to do it. Another way to think about it: Have to = Required. You will be punished if you do not do something. Should = Obligation. You have a responsibility, to yourself or others, to do something.

When you "should do" something it means you or other people may think you are going to do something and it would be good to, but when you "have to" do something it's a necessity and there's really no other right option.