Other mistake: When Professor Potts bought Chitty from the Junkman and brings the car into the garage, the body of the car is charred or damaged somehow from the crash and fire. But the leather seats, dashboard and wooden steering wheel are completely unharmed.
Other mistake: When Chitty Chitty Bang Bang passes the train on the first drive out there is no smoke coming from the train's smoke stack, meaning the train isn't running. Yet it is definitely moving.
Other mistake: In the scene where everyone is going to get away from Vulgaria in the car, Grandpa scrambles in after them. You can distinctly see a Vulgarian soldier give him a shove to get him in the car, then do his "Run away." thing.