There's no difference in the way I've ever heard them used, I regularly say both, but I probably say 'elementary school' more often than 'grade school'. The only difference between states, and actually even in the same city, is how many grades an elementary school has. The elementary school I went to has grades 1-6. Some other elementary schools in the same city have grades 1-8.
(not a teacher, just a language lover)
Student or Learner