Depending on what it is you hope to see, I might vote for neither.
Neuschwanstein draws 1.6 million visitor per year. To get 7,000 visitors per day through the place, tours leave every 5 minutes and last only 30 minutes. While most palaces in Germany get mostly German visitors, N'stein and the other kings castles are a mecca mostly for international tourists - well more than half are from China, the USA, and other foreign countries. I recently read this comment in German from one of N'stein's tour guides in Germany's DW online.
Schlossführer Patrick Korb (32) hat festgestellt, dass Besucher aus unterschiedlichen Erdteilen auch unterschiedliche Erwartungen hegen. „Amerikaner haben oft das Bild von der perfekten Mittelalterburg im Kopf. Wenn man ihnen dann sagt, dass das Schloss zu einer Zeit gebaut wurde, als in Chicago schon die ersten Hochhäuser entstanden, wollen sie das manchmal kaum glauben.
(Rough translation - Americans, like other foreigners, often base a visit to N'stein on misunderstandings - Americans often picture N'stein as a medieval castle and often do not believe guides like Korb when they're told it was built around the same time as Chicago's first skyscrapers.) N'stein was given a fake castle exterior but isn't remotely a castle - it was built as a royal residence.
This DW article in English explains the uniquely tourist-heavy N'stein experience.
Rothenburg is similar. The math tells the story - 2.5 million visitors per year in a town of 10,000 locals who serve them food, sell them trinkets and clean their rooms. Half the overnight guests are foreigners.
It's almost as though there's some conspiracy afoot that conspires to shuttle all Americans and other foreign tourists to Rothenburg and the King's castles. Yet Germany has hundreds of real castles and modern palaces you can visit... If you're looking for a faux castle - or destinations where Germans are outnumbered, and English is language of menus and business - then Rothenburg and the King's Castles may suit you. OTOH Munich itself has a fabulous palace you might have overlooked - Nymphenburg, King Ludwig's birthplace. It includes a main palace, satellite palaces, a carriage museum, a porcelain museum, and gardens. You'll find some international tourists of course, but their numbers are much smaller and you won't be rushed through in a milking line. And there are genuine medieval castles you might visit as well in Bavaria, including Pappenheim, Coburg, Burghausen, and Harburg; a trip to Salzburg and nearby Werfen would provide access to other castles including the impressive Hohenwerfen (film site for the late-60's move "Where Eagles Dare") where there's also a falconry show.
