Rothenburg is one of those places that has a good and a bad version of itself depending almost entirely on timing, and the difference is whether you are there with the day-trip crowds or without them.
The worst time is the peak of summer (July and August), when the Marktplatz and the Plönlein can feel like airport departures. The Christmas market season (late November through December) is the most crowded of all but also the most beautiful — the town is decorated, the market stalls sell hot Glühwein and regional crafts, and the medieval setting makes it arguably the finest Christmas market backdrop in Germany. If you come at Christmas, book accommodation months ahead and come on a weekday.
May and June are the best months: the weather is warm, the crowds are manageable, and the Meistertrunk festival (Whitsun weekend) fills the old town with locals in period costume rather than tourist buses. September and October are excellent too — cooler, quieter, and lit well for photography. April can still be cold but is very uncrowded.
The single most important timing advice is about the time of day rather than the season: arrive early in the morning (before 9am) or plan to stay overnight. The old town between 7am and 9am, or after 7pm when the coach tours have gone, is a different place entirely — the lanes quiet, the Plönlein empty, the walls walkable in something close to peace.