How close to being on a straight line between Rothenburg and Stuttgart do you want. The first place that comes to my mind is Schwäbish Hall. I've never been there myself, but I know it's a popular tourist spot. Closer to Rothenburg, there are a number of other towns on the Romantic Road - Bad Mergentheim (home of the castle of the Teutonic Order of Knights), Weikersheim (with it's palace and gardens), and the walled towns of Dinkelsbühl and Nördlingen - the last two maybe too much like Rothenburg. A little more out of the straight line is Würzburg, with it's Residenz and Marienberg Fortress.
As mentioned above, Tübingen is a pretty University town, but hardly between Rothenburg and Stuttgart. Near Stuttgart, Hechingen is home to the Burg Hohenzollern, an impressive "castle", although built in the mid 19th century, about the same time as Neuschwanstein.