beach - This is more often used to speak of a place for recreation and to have a good time.
shore - This is more likely to simply refer to the coastline.
In New Jersey, I believe the common phrase is "the shore". I think they go to the "shore" and the "beach" in New Jersey. They refer to the New Jersey coastline as the "New Jersey Shore" if I remember correctly. Anyway, Google should confirm this (or not).