Haven't been on AF in a couple of months now (to JFK, maybe 2 or 3 months ago) but my last LH experience is from last week (EWR) and another one two weeks earlier (ORD)

Generally: (1) both are SUPERB. As long as you don't go on British Airways all European First Class products are fine. And (2) all comments below are comparing the Air France 777 with the Lufthansa flight, the AF A380 being not as good

Cabin feeling: The AF 77W has only four rows, and feels slightly more claustrophobic - but not on a level that it's really disturbing. The Lufthansa A380 feels much roomier, and the B747-8i even more so.The B747-8i feels nice because the First Class cabin is in the nose of the plane
Seat quality: You won't hear me say anything bad about either. Both are fantastic. Enough storage, table large enough,comfortable in any position. Subjectively: I prefer the Air France one for sitting, the Lufthansa one for sleeping, but that may also have to do with the following point:
Privacy of the seat: Hmmm... for me that is the weak point of both seats and I prefer the setup on for instance LX. To get good privacy on AF, you need to pull the curtains, which makes it then feel more claustrophobic, a bit like a field hospital. The Lufthansa seat has the "shield" or "wall" around the back and side of your seat,which is slightly better. But when it's down, you are pretty much exposed to the rest of the cabin.
Flight attendants: Again, both are great. I find the Air France ones more elegant and "First Classy", the Lufthansa ones more attentive and proactive (not to say that the AF ones are not, but LH usually better). A question of personal preference, really. Also, it differs from one crew to the next. Consistency usually pretty good across crews in First on both carriers
Catering: Again, personal preference. But actually, I find the LH catering to be more achieved than the one on Air France. Also more choice on LH. Flights from the US on Air France have a not so great reputation for catering, but Lufthansa is actually good (if not as excellent as leaving from Europe). Both have dine-on-demand, and both have a variety of things they can serve you outside of the main meal services if you feel peckish. Lufthansa choices somewhat larger and more refined for that snack service.
IFE: (I usually don't care.) Again, both have large screens, although I think the AF one is somewhat larger? What makes me prefer the AF system is the choice of more artsy and French movies, but that is just my taste for movies. People that are into the typical stuff served by the US airlines (a spider mutes into a giant spaceship, Manhattan is in ruins after 3 minutes and the longest verbal exchange in the film is "My revenge will be sweet" replied by "Hmmpff") will also find something that doesn't awake their brains too much
Internet/WiFi on board: the only true difference. If it matters to you, then don't take Air France, they don't have it. Lufthansa has it on all longhaul planes (and soon all shorthaul)
Amenities: Lufthansa has the nicer Pajamas, Air France the nicer amenity kit. But again, both are great.
Ground experience: "Miles ahead" for AF is an exaggeration. It depends:

• leaving from JFK, both airlines will escort you through security, and bring you to their lounges. Lufthansa has an entire floor of its lounge for First Class à-la-carte dining and a large buffet, the quality is okay-ish. Air France has a dedicated section, I've missed it last time I went through JFK - check the posts here on the forum, not sure iwhat the dining experience is there.
• Transiting in either direction is better in Paris, no doubt. You get picked up at the aircraft door and a limousine drives you to the Premiere lounge and then to your connecting flight. But you have to go through security when arriving from the US.
• Transiting USA-Europe in Frankfurt, coming from the US, you typically arrive at the Z gates, and the First Class lounge is right there after passport control, no need to go through security. For your connecting flight, if it leaves from an open stand, they'll drive you. Otherwise another short walk in the same concourse, just one level down. The walks are not long, but definitely more walking than in the case of Air France where they drive you
• Transiting Europe-USA in Frankfurt, again, depending where you come from it's just a short walk to the First Class lounge and then on to your US-bound flight. Basically the US flights leave on the upper floor of the concourse from where the European flights leave.
• Arriving Europe better in Paris: they usher you through passport control, which can be a very very messy place in Paris. In Frankfurt, no such assistance, but usually not such a mess either. Lufthansa is much better with priority luggage than Air France
• Leaving Europe better in Frankfurt, as the First Class Terminal is a full terminal building in its own right, entirely dedicated to First Class pax. You can get out of your car in front, security is right behind, and the lounge (the entire terminal is actually a lounge) right behind that. From car to lounge it's about 15 meters. In Paris, other than getting to the actual place where the Premiere staff awaits you being an absolute chaos (not AF's fault), you still have to cross the full Terminal 2E. You do so with an escort, but it's a walk which is about as long as the walk from the First Class lounge to the gates in Frankfurt.
• Arriving USA the same. Both airlines have staff that escort you through immigration in JFK Terminal 1.
• All in all... At JFK, slightly prefer Lufthansa, terminating and transfering in Europe I prefer Air France, originating in Europe I prefer Lufthansa