You're probably going to get a lot of responses.
In very simple terms, Hogarthian diving (which I believe is named after Bill Hogarth Main) strives for simplicity, eliminating everything that isn't absolutely necessary. "Hog" divers typically use a backplate and wing (bp/w) with a single piece of webbing for a harness.
DIR ("Doing it Right") is a dive philosophy that embraces a particular hogarthian equipment configuration as a part (and certainly not the entirety) of an overall approach to diving that focuses on team diving and skill mastery. DIR divers have done some very impressive dives with an outstanding safety record, notably extensive cave explorations in Florida. The DIR system works for any kind of diving, but some feel it is overkill for strictly recreational divers. Others strongly disagree.