being american, i would use cemetery as the "official" word (going to a funeral) and graveyard referring to an older, no longerused cemetary...
it would be interesting, to read how the british would use both terms.
Grave and yard (and hence graveyard) come from Old English words. Cemetery is from the Greek kimitirion (κοιμητήριον), which means something like "place for sleeping". English has a lot of instances like this where two words, one of Classical origin and one of Saxon origin, have the same meaning. Generally the Saxon words seem to have a more common or "earthy" quality to them, and the Classical words tend to seem a little more "official" or scholarly.
I would agree with river, we have graveyards, more commonly called "churchyards" with a church on site and "The Cemetary" is a place, usually in a town, where people are buried in large numbers. This has no church on site, although some have a small chapel. This site is usually maintained by the local authority not a specific religion.
The church/no-church distinction is the only one I know of. Of course, there are more graveyards in Euope than in the States, and we have more cemeteries than graveyards. In addition, at least in the States, more and more churches are creating columbariums to hold the cremated remains of members. Some are quite elaborate with statuary, etc., but I know of two that are in the area outdoors where good-weather coffee-hours are held, with benches right along the walls where the niches are labeled.
As long as we're being earthy, let's not overlook the wonderful option boneyard. It's a little on the figurative side, though in another sense it's as literal as a word can be.
We have monuments in cemeteries and tombstones (or headstones) in graveyards-- for some reason gravestone is considered a little blunt. Odd, since tombs are enclosed structures, not like bare, open graves where we Americans erect "tombstones."