I have flown both airlines on transatlantic routes (in coach) and I would go with BA. I like them better, even though LH is pretty good, too. Check out what kind of planes you'll be on, BA's 747-400 and 777's have PTVs in coach (if you like that), LH does not have them. BA's 747-200 do not have PTVs and are similar to LH's. If you'd like to fly on an A340 "retrojet", you have to go with LH. They are very quiet, but other than that there is nothing exciting about these planes. If you are tall you may find it hard to sit upright in a window seat. The combination of yellow and grey that LH uses for their interiors does not do it for me, either. The overall level of service on LH will be OK, but I would rate BA quite a bit better.
Either way, you can probably go with whatever airline offers better fares and more convenient connections. FRA has a GREAT observation deck, you can get up close and personal with the planes. But also keep in mind that LH captains have been on strike recently. It caused a lot of cancellations on their domestic routes.