The two train stations (regional and long-distance) are right next to
each other, one below the other. No advantage.
Not true, the Regionalbhf is under the building right across the access road from Terminal 1. When you leave the Regionalbhf you come up stairs or escalators into the hall of Terminal 1.
The Fernbahnhof is farther out, on the other side of the Autobahn from the Regionalbhf, although you don't realize it because you are going through an enclosed walkway.
It's a little faster to checkin coming in to the Regionalbhf, but what really matters is what kind of train you want to come to the airport on. The Fernbahnhof is the Bahnhof for the long distance trains (ICEs and ICs). The Regionalbhf is for local trains (REs, RBs, and the S-Bahn).
I would think that coming in from Heidelberg, you would want to come in on an ICE to the Fernbahnhof.