I do both, because our site is small and I don't know who's going to get around to it first. Once you flag it, it's in the mod queue regardless of the open/close status.
Voting to close can close the question, which at least stops other users from being able to interact with it. Also, once it's closed, we can vote to delete the question (at a reputation >= 2k).
Mods are supposed to be "exception handlers," but we the community are supposed to handle day-to-day operations.
Our mods aren't around on a daily basis, which means also that nobody is checking the mod queue on a daily basis. If we can close and delete the questions on our own without their intervention, great. If they remove the questions before we can close and delete them, that's also great.
So again, I do both, and whoever gets to it first gets the question removed, which is ultimately the point.