I think of a cavern as much larger than a cave and with stalagmites and stalactites. I think of a cave as something low ceilinged, claustrophobic, dark and potentially much more dangerous than a cavern. For some reason a cave seems more wild and natural and a cavern comes across as already partially "tamed" by humans, perhaps miners or explorers.
There really isn't a difference. The terms are used interchangeably but historically "cavern" has been used to make a cave more glamorous. Consider this, Carlsbad Caverns National Park was once Carlsbad Caves National Monument.