I would say that there is little advantage to purchasing regional tickets in advance, online, but buying them at the counter is definitely not a good idea.
First, there is no ticket counter in St. Goar, only an automat, and I doubt that Rothenburg odT is purchasable from that automat. You could purchase a ticket for two to Mainz (€20,60) and then, at the counter in Mianz, purchase a ticket for two to Rothenburg (€71,60). That's €92,20 for both of you.
If you were somewhere else in Germany with a ticket counter before you got to St. Goar, you could just buy two through regional tickets from St. Goar to Rothenburg for €82,80. If the counter worker was extra nice, he might sell you a regional ticket for two from St. Goar to Kahl am Main (€43,80) and Bayern-Ticket (€30 counter price) to use from Kahl to Rothenburg, €73,80 total.
Travel by regional train from St. Goar to Rothenburg will take 5½ hrs. However, if you go online, to the Bahn website, and purchase a Dauer-Spezial Familie ticket from St. Goar to Rothenburg, it might only cost you €49 (for two). That is for a 4½ hour trip with an ICE leg from Mainz to Würzburg.
So, what will it be - regional train tickets at the counter for €74-€92 or express trains tickets, in advance, online for €49? The Dauer-Spezial ticket does commit you to a specific ICE from Mainz to Würzburg, but look how much you save.
For Nürnberg to Munich, the price of a regional ticket for two at the counter is €61,60. Or you can do the same trip on the same trains for €28 with a Bayern-Ticket purchased from an automat.