I'm afraid I have to strongly disagree with Jackie on this suggestion. If you have been using your title in correspondence (internal or external) and you change it yourself without company approval, you are setting yourself up at the very least for chastising and at the worst, potential disciplinary action. Many companies would not take kindly to employees giving themselves a "promotion" (lateral or otherwise) and as you would have had before and after correspondence, it would indicate that it was a deliberate change to your signature.

I would talk to your HR folks and boss. I'd make an appointment, go in with a list of your current responsibilities and both job descriptions and then make an argument for your title to classify your job appropriately. Since there is no change in salary range, I would push hard for the accuracy and question the "why" if they refused. 

Let the title request stand on its own merits though. Don't try to justify it by saying everybody else has the other title unless you can objectively prove that everyone's job responsibilities are identical.