The term Spaceship most commonly means largely the same thing as Spacecraft, but sometimes with the added connotations of being "ship"-like.
Sometimes Spaceship is used informally (and arguably incorrectly) to mean UFO.
Game and Story Use
- The "Spaceship" vs "Spacecraft" terminology may actually have meaning in your setting. You might choose the term that better fits your setting, or have different space agencies use differing terms.
- In a military sci-fi story, the main astronauts may be Space Marines whose organization grew out of planetary Marines, who in turn grew out of the Navy. In such a setting, calling your spacecraft "ships" is a natural progression.
- By contrast, if your primary source of astronauts are coming from an organization birthed out of the Airforce, then spacecraft (derived from aircraft) might be more appropriate. Heck, you might even go with Space Plane if your setting features lots of Old School Dogfighting.
- The term "Spaceship" may also crop up because of the common speculative fiction trope known as Space Is An Ocean. If that's how space works in your setting (in the real world, space does not work that way), then it makes sense to extend the naval metaphors.