Limekiln is the state campground, Kirk Creek is the forest service campground.
They are less than 2 miles from each other. I have stayed at Kirk Creek twice, there is a hiker/biker site, the setting is lovely. It just has cold water bathrooms, no showers or other amenities. Limekiln is tucked away on the east side of the road in an indentation in the coast, kind of a land-locked cove, and it's not so appealing looking - looks kind of dark. The Pacific COast guidebook says it has showers & more amenities, but I never went in there to check.
Kirk Creek is on a point that sticks out into the ocean a little, and is very sunny. It's a spectacular sunset spot.
The h/b sites are away from the car sites, but it's a quick walk to them. There's a little trail out of the h/b sites that takes you to the cliff edge, and down to the beach if you want.
One of the times I stayed at Kirk Creek there was a sign on the water spigot saying to boil water, so be prepared with a way to treat water or extra gas for boiling, or call the state park service ahead of time for info. I guess worst case you could ride back up to Limekiln and either stay there or acquire water.