You could say one is specific to New Jersey—and colloquially, that would be correct. But technically, people in New Jersey can go to the beach, and there are shores even outside the Garden State.
"Shore" is a generic term for the place where land meets water. Any land that directly borders a big wet area is a shore—be it sandy, rocky or cliff-like. The "beach," meanwhile, refers to the area at or along the shore that is characterized by sand or small pebbles conducive to lounging and castle-building. When it comes to summer vacations along the Atlantic or Pacific, these are often interchangable, but that's not always the case: Large rivers have shores and sandbars are beaches.
So when New Jerseyans head "down the shore," that is true, but they're probably more interested in the beach.
Oh, and a coast is a type of shore that borders only oceans—not other bodies of water.