"Spaceplane" is a fairly well-defined term. Everyone basically agrees that a spaceplane is a vehicle which is capable of aerodynamically-controlled flight in the atmosphere but can also operate above the Karman line.
"Spaceship" and "spacecraft", on the other hand, are a little less neatly defined. NASA seems to prefer the latter term, but I have seen several media sources referring to a variety of vehicles as spaceships, including the Dragon V2 and others.
Now, I no there's really no rhyme or reason to this; it's just a matter of preference/convention. But if we were going to officially distinguish between the two, how would we go about doing it? Would spaceships be a subset, a distinct type of spacecraft? Do we say that a spaceship is manned and a spacecraft is unmanned? Or do we choose more stringent requirements for spaceships, like having ascent engines, maneuvering/transfer engines, and extended life support?
Other suggestions?