Depending on timings, head due west to Uberlingen, gorgeous little lakeside town, loads of restaurants/bars. Then head further west and pick up the B500 (legendary twisty) heading North to Baden Baden. Then perhaps Reims then up to Calais?
If you're going from Stuttgart to Neuschwanstein why not take a little detour and have a wander around Dachau? Not exactly the chirpiest place in the world but very, very, interesting, and moving.