I went from Nördlingen to Rothenburg by bus in 2007, following the Romantic Road via Dinkelsbühl and Fuchtwangen. I used buses that can be found on the Bahn website. Actually, I only went from Nördlingen as far as Feuchtwangen the first day, because I spent the first half the day seeing Nördlingen, then took the bus after lunch to Dinkelsbühl, spent several hours, then went by bus to Feuchtwangen for the night. The next day I took a bus mid-morning from Feuchtwangen to Rothenburg via, as SLA noted, a short change in Dombühl , and arrived in Rothenburg for lunch at the station. I didn't spent time in Rothenburg because I had already been there twice. After lunch I went on by bus to Weikersheim, also on the RR.
If you go from Rothenburg to Nördingen by bus, you can continue down the Romantic Road on with the train via Harburg, Donauwörth, Augsburg, to Landsburg am Lech. From Landsburg to Füssen there are again buses that follow the Romantic Road.
The trouble with using the Romantic Road Coach is that the Coach only makes one run a day, leaving Rothenburg at 10 after 1 PM and arriving in Nördlingen at 3:25 PM. It stops in Feuchtwangen for 15 minutes and in Dinkelsbühl for 30 minutes. Thirty minutes is not enough time, in my opinion, to adequately see Dinkelsbühl. Nördlingen would be a dandy place to spend the night, then go on towards Füssen by train and bus. If you continue on the RR Coach to Augsburg, arriving at 5:40 PM, you only have 30 minutes in Nördlingen, which again, in my opinion, is insufficient time to see the town.
Traveling on the Romantic Road is only worthwhile if you spend time in the towns. The Road is nothing special, just a two lane country road with a slow speed limit, clogged with freight trucks and farm equipment.