Speaking generally here, I think the best approach tends to be to address it head-on. Include in the text of the question an appropriate statement like "I'm aware of related question X, which addresses a related problem but does not actually answer my question because...". That is, spell out the differences to make it easy for the would-be close voters to not go with their initial reaction.
If you are aware of this related question prior to posting the question, include it at time of asking. If you were unaware of it, and someone points out it out as a potential duplicate target in the comments, edit the question at that point accordingly and reply to the comment as appropriate. If you're too slow to respond, the question might be closed in the meantime. In that case, edit your question at that point and make the case that it wasn't a duplicate.
That said, what constitutes a duplicate is somewhat fuzzy and a perfect duplicate does not have to be found for closure to be appropriate. In particular, "not case specific" is not in itself a good argument if the duplicate target is more general, and the answers would apply also to your more specific case.