High fantasy can be a bit controversial to define, and it often gets mixed up with epic fantasy. If you want the What's The Fox Hat definition, then a high fantasy is a fantasy set somewhere that isn't earth, but on a whole new world (not in space, that would probably be sci-fi). If the story in that world is massive empires and kingdoms clashing it's probably more epic fantasy. If it's more of a persons journey around this fantasy setting it's probably high fantasy.
Lord of the Rings is a journey of a small fellowship, but it's against a backdrop of a massive war. The Hobbit is a journey of Bilbao and his mates, but without Mordor and elves having a scrap. So Lord of the Rings is more epic than the high fantasy of The Hobbit. But to be honest we would be happy with it being classed as both. Clear? As mud.