Hurry is the general verb.
"Hurry, hurry, hurry!" <--- Be quick (run quickly or think fast).
"Come quickly! hurry!" <--- Note you haven't set out yet, but you are being asked to hurry as you make your way to the person calling you.
"Hurry up" basically means "hurry more". Since it only makes sense to hurry more if you are already doing something, "hurry up" is more task-specific than "hurry".
"Haven't you finished your essay yet? Hurry up and get it done!" <--- You are already writing, but the person you're talking to doesn't think you're doing it quickly enough.
Student or Learner