For my first license in Tennessee I only had to do an eye exam. I'm heavily nearsighted, but with most states you can take the exam with corrective lenses, on condition that you get a restriction put on your license that requires you by law to wear glasses/contacts while driving.
They also had a section to write any physical or mental disability that "may interfere with your driving ability." I have an Asperger's diagnosis, but I didn't write it down because it doesn't and shouldn't affect my ability to drive in any way.
The first mistake you made was mentioning Asperger's in the first place. Now given, they did put you through a medical exam, but since you got a pass grade from that, you should have been ok with not mentioning it. The review is only supposed to deal with conditions that may interfere with one's driving ability; however, the fact that you voluntarily disclosed your diagnosis before the committee implies that it DOES affect your driving ability, even though you and I know it doesnt.
Given, that was a mistake, but everyone makes mistakes. Now here is my advice to rectify your mistake:
First, get legal counsel. Even if you don't need a lawyer to represent you, it helps to have one. Their job is to learn about the convoluted labryinth that is law and invoke it to the client's benefit; thus, they will know the appeal process better than you will and can help you in that respect.
Second, get an expert opinion. If you have a psychiatrist that you visit, talk to them about it. They should be able to present medical evidence that you are indeed fit to drive, which you can then review before the committee.
Finally, prepare yourself mentally for the task of driving, because I can forsee one of the possible outcomes of your appeal being a practical exam to ascertain your ability and whether or not your mental state affects it. If it comes to that, make sure you know what you're doing. If you blow them out of the water with your superb driving skills without faltering, there is no way in legal hell they can deny you a license. But you'll probably be under more scrutiny than the average applicant, so if you'll pardon my language, don't f**k it up.
The law is on your side. Like Tory said, states can't deny licenses unless the person is an illegal immigrant, has previous traffic offenses resulting in a suspended license, is physically/mentally unfit to drive, or other reasonable cause. The phsycially/mentally unfit clause is usually reserved for severe cases, i.e. amputation of limbs, schizophrenia or other psychosis - i.e. when the person endangers himself or others by attempting to drive. Aspergers does not fall into this category, therefore with a proper appeal they should give you the go-ahead.