Buy a car in another state, pay taxes twice?

Living on Missouri I bought a car since my previous was totaled by someone and then 1 month later moved to Illinois or ILL & ANNOYED. I paid Mo. tax and was told I had to pay IL. tax since I brought it into IL. within 6 months of purchase. Well, my wife left without getting the IL. plates to talk to me about it. I simply waited 3-4 months and went back to the DMV and they did not say a word about it. Who can figure?