Train from Frankfurt Airport to Rothenburg ob der Tauber

Yes, you have done a super job of figuring it out - the Regionalbahnhof is only for local trains (for instance, those in the Rhein-Main Verkehrsbund, such as trains to Wiesbaden). Typically their number starts with an S or a RB. The Fernbahnhof (fern = far) is for trains that travel a long distance from the station (ICE, IC numbers). Rothenburg is about 2 and a half hours from Frankfurt, not just because it is far (it is) but also because it is in a rail cul-de-sac necessitating changes of trains.

Lavandula

If the first train is an ICE train, it will leave from the Fernbahnhof. Right now, a Flexpreis Ticket (cost if you just walked up and bought a ticket), is 78.40 EUR per person. Takes about 3 hours with 2 train changes at Wuerzburg and Steinach.

If you go to the DB site and click the box "Local transport only" under the "Mode of Transport" column. Then you will see the route using only regional and S-Bahn trains that leave from Regionalbahnhof. These cost 49 EUR for the first person and 10 EUR for each additional person. It is called a QdL ticket and is good after 9 am week days or anytime weekends. There is one extra connection before you get to Wuerzburg, either at Frankfurt Hbf or Hanau, and the travel time is just under 4 hours.
Hint: deselect buses in the mode of transport. Otherwise the results may suggest a bus to Frankfurt Suedbahnhof. The bus, of course, will be from outside, and the cost is not part of the rail cost.

"When searching for information about the journey, several sites indicate that we would need to take a train from the Regionalbahnhof station at the airport."

By default, the DB site only shows the fastest train journeys - so if the faster long-distance trains are possible for your route, then journeys that include such trains (ICE, IC, EC, etc.) are probably what turns up on a search.

But you do have the choice of regional trains only. Sam gave instructions if you WANT to go regional only. Those trains depart from FRA's Regionalbahnhof.

Something you might not know is that there is a month-long pass for regional-only train travel - the Deutschland Ticket. €58/person/month. Fantastic deal if you plan to use trains after Rothenburg. Also good for Subways, Buses, and Trams throughout Germany. You would not need the QdL ticket for FRA > Rothenburg if you get this pass:

https://www.seat61.com/train-travel-in-germany.htm#Deutschlandticket

Wow, thanks everyone!

So to summarize, you can get from the airport to Rothenburg using EITHER station! The regionals are cheaper and slower (including more transfers), the 'far' trains are faster and more expensive.

Given that we will be jet-lagged, taking the faster, fewer-transfer trains sounds like the best option - though pricey!

I posted separately about renting a car. I'm now deep in the weeds of pursuing that option also.

Thanks again!

You say one minute that you will be too jetjagged to take 1 extra train change, but now you want rent a car, probably sleep deprived and drive 2 hours through unfamiliar territory without endangering yourselves and others. Hmm.

Frankfurt Flughafen Regionalbahnhof, an - Fernbahnhof are officially the same station. Regionalbahnhof is tracks 1-3, and Fernbahnhof tracks 4-7. So it does not matter which one you pick when planning.
DB has divided it in sectors in their planners (similar to you have Berlin Hbf Tief, and the different sections of Munich Hbf) in order to be able to take in to account that the walking distance between 1-3 and 4-7 is not trivial.