Lufthansa train service

Lufthansa do not run any trains, so you will get a seat on a regular DB train. I've done it once, even if it was many years ago. But it was easy and smooth, there was a check in counter at the station where you could check in for your trip and get a boarding pass that also acted as a train ticket. But the routines might have changed since then.

German Rail has a program, Rail&Fly, in cooperation with several airlines, including Lufthansa, where one of your flight coupons is actually an open rail ticket from any train station in Germany to any airport with Lufthansa flights or visa versa. It cost 33€ (?) per connection, and allows you take any train, including ICEs, at any time on the day of your flight, or one day before or after.

But I believe Lufthansa does run some dedicated trains, called Luftahansa Express to and from airports.
It might only be to and from Cologne, but I think there are others.

Yes, rail&fly and LH Express Rail are different concepts, but neither uses a "dedicated train", you will travel by regular ICE. R&F is a cheap train ticket that you may book in addition to your flight, LH Ex is technically the first or last leg of your flight and you will get a boarding pass for it. It's bookable from/to stations that have an IATA code. The advantage is that it offers way more connections to towns with minor airports. There are, I believe, four or five flights between FRA and Nürnberg a day but there is at least one train per hour. Another advantage is that if your train is delayed your flight will be protected, which is not the case with R&F.

Please note: There is no check in at the Nürnberg station (as used to be in the initial phase of the project). You have to check in online; you will get a reserved seat and you will take your luggage with you. In FRA, you can check in your luggage at the AirRail terminal.

Lufthansa did run their own trains until 1993, https://en.wikipedia.org/wiki/Lufthansa_Airport_Express .

But now you get a seat on a regular ICE.

Yep, I used it this past Sept. I checked in via the Lufthansa App and downloaded the boarding passes. The first boarding pass was for the train and I just showed it when they checked tickets. Super easy and came with seat reservations.

DJ

Another advantage is that if your train is delayed your flight will be
protected, which is not the case with R&F.

I don't know that I would ever use Rail&Fly to get to an airport on my way home (i.e., R&F, then flight). I usually stay close enough to the airport (like in Mainz for FRA, Freising for MUC) so that the fare in is less than R&F's fare. Where I would use R&F is for my arrival in Germany, where I often travel farther (several hours) and I want a low fare that I won't lose if my flight is late arriving.

On the other hand, for 2 of my last 3 arrivals at FRA, I've been able to use a combination of local tickets and/or regional passes so that I was not specific train constrained and my fare was lower than the R&F fare.