Outside of diff: Graphite grease Inside of diff: Silicon oil Lock the diff: Silicon ear plugs.
So greasing the outside of the diff with make things smoother. Greasing and oiling anything on your drive train will make things smoother and faster as less friction. So drive shafts ends, bearings, diff outers and drive hubs.
Inside the diff is more about corner rate. Generally the thinner the oil the looser the diff the tighter the turn as when turning, right for example, the inside will travel less of a distance than the outside wheel - generally meaning the wheels are moving independently easier. The stiffer the diff the less turn you get. Locked diff will reduce the turn as both wheels are forced to travel at the same rate but acceleration is increased. Can scrub your tires up though. BUT - Good on crawlers where you need the traction over turning on certain obstacles.