I take both these airlines in FC quite frequently (2-3 return FC trips/month on average, 30% on LH, 20% on AF, rest mainly on LX), so I'll try to maybe paint a picture that goes beyond the transfer experience.My last LAX-Europe was in August on AF, and my last LAX-Europe on LH was about 1.5 years ago, so the LAX-specific details are a bit from distant memory.

Personally, it isn't as clear-cut for me as to some other posters. In a nutshell:

1. If the transfer experience is what matters to you: take CDG over FRA (it's not quite the disaster others make it sound, but CDG is 100% great. More below)
2. If you care about in-flight connectivity, take Lufthansa. Air France is 95% certain not to offer it
3. For the rest, it's a matter of taste
4. Depending on where you fly to afterwards: also consider FC on Swiss through Zurich, or Lufthansa on the A380 through Munich. Very pleasant transfer experience as well.

Departure: same same. Both airlines have dedicated CKI counters, no wait, both will escort you through securtiy to the lounge and pick you up for your choice of early or late boarding, usually with a dedicated escort per pax. Air France uses the oneworld/operated by Qantas First Lounge, with order-to-the-table service. Nice. Lufthansa doesn't have its own lounge either, I forgot what it was called (as far as I remember it wasn't the normal Star Alliance/run by NZ lounge). Similar services.
Cabin overall: matter of taste. Air France with only four seats has the smaller cabin. Without the curtains closed it feels more "packed" than the LH Jumbo Jet, the seat across the aisle is really close and the cabin is sort of closed in in front and in the back. There is a 50% likelihood you end up in one of the middle seats, which I find dreadful (I remember two recent CDG-Asia-CDG flights where I was in the middle seat, curtain closed and the divider wall to the neighbour raised. Freaked me out, I managed to sleep 2.5 hours. But that's me). So in the end, whether it feels more private or more claustrophobic depends on you. Lufthansa has six seats by the window and only one pair in the middle, so your chances to have a single seat are much higher. The cabin doesn't feel cramped, as neighbouring seats are further away thanks to the larger real estate in the nose section of the Boeing 747. Less provate or less cluastrophobic? Your call. In terms of design/colours Air France is nicer.
Seats: matter of taste. Both are large and great for sleeping. Lufthansa has a bigger table, and I usually find it easier to find the perfect seat/table config for eating and working, the AF one comes a bit close to where I am sitting, unless I put the seat almost upright which isn't comfy either. Lufthansa seat better in lounge position in my mind. Bedding is great with both, although the Air France comforter is slightly narrower (but not as ridiculous as the BA one which basically always leaves a part of you uncovered), which sometimes bothers me. Back to privacy: Air France has the curtains, which - at least for me - is fine when in a window seat, but claustrophobic when in the middle seat. Reminds me of a World War I field hospital. But you have your own space. Lufthansa: I don't like how they have mounted the TV screen on pylons, it gives the seat such an unstructured feel, making it less private. But it all gets better when you raise your privacy screen, which covers your back and side. All in all you are not as enclosed as in the AF seat. More private or more claustrophobic? Your choice
IFE: matter of taste. If you like French movies, of course AF. But Lufthansa has live TV, and I remember fondly how I watched some Football ("Soccer") World championship matches above the Atlantic.
Connectivity: Lufthansa. Very simply, Air France has installed WiFi on very few of its 777 planes, so better start with the assumption that you won't get it. So if it matters to you, don't take AF.
Catering: Lufthansa, esp on flights ex-USA. Again, matter of taste and if you like a particular dish or not, but generally the Lufthansa preparation, taste and presentation are better. Lufthansa on some routes has more choic (not sur eabout LA, but I remember other ex-USA routes where AF hardly has any choice and breakfast is a joke. But that is more East coast with shorter flight times). There are details: AF has better desserts, the LH salad is fresher, AF breakfasts are typically overvcooked, Lufthansa bread more choice but sometimes more stale. Where they are the same: both have a caviar service, although the LH appears more "copious" (they basically come round with the trolley loaded with caviar boxes and you can take as much as you like, chosing your own accompagniements. On AF, I remember having little hors d'oeuvres that had caviar on them). If you like caviar, that may matter. Both have dine on demand.
In-flight service: same same. Both are top notch, really. The cliché of the more elegant French service vs. the more robotic German one doesn't work anymore. Both are extremely attentionate, responsive, courteous, on their toes. It really depends on the crew/the individual FA you have. FA/pax ratio are the same. 1 FA per 4 pax service (=1 on AF, 2 on LH), and one in the kitchen during meal times.
Arrivals services: Air France. Simply because they offer 100% plane>lounge>plane service. Easy to explain: when arriving in CDG, you get picked up by a personal escort who will drive you to the Premiere lounge, where you go through passport/security, and then drive you to your next plane. Bear in mind that if coming in from LAX you will have to go through security in CDG - but not so in FRA, as they have the SSS (single stop security) with a couple of countries, among them the US.
Lufthansa in FRA: it depends. If your plane arrives at a tarmac position, they'll also pick you up by car and drive you to the arrivals area, from where it's a short walk to the First Class Lounge. But it's unlikely for an incoming 747 from LAX to have an open stand. Otherwise you have to make your own way from the arrivals gate to the loune. But frankly, that isn't such a big deal: if you arrive in A/Z, it's a short walk, you go through passports which usually is a quick if you have a European passport, and the First Class Lounge is right after passport control. If your connecting flight leaves from a bus gate, they'll drive you with a limo straight to the plane. If it's on a jetway in the same concourse, you'll walk there (short walk). If by chance you arrive in the B concourse, it's similar, as there also is a First Class Lounge right there. But US flights typically arrive in A/Z I think.There are variations of this, depending if your connecting flight is to a Schengen destination or not, etc. You'll never have to take a bus (that would always be a limo), but there may be walking involved.

But this is where I would throw in the alternatives of MUC and ZRH, if that works with your routing:

• MUC: First Class flight on the A380, very pleasant (quieter and more spacious than 747-8). Upon arrival, walk a really short walk on the same floor to the First Class Lounge, without having to go through security. Distance-wise this is the same as walking between the Premiere lounge in Paris and a gate in the 2E-K concourse, where even AF wouldn’t drive you. Is straight from gate to First Class Lounge, there is a passport counter in the lounge, so you're through in 10 seconds. If your connecting flight is in the same concourse, walk there (really short again). If it's from the other concourse or an open stand, they'll drive you with a limo. There is a First Class Lounge in every concourse. The a-la-carte dining isn't as good as in CDG, but all the rest is actually more pleasant, especially the FCL in Satellite 2 (K and L gates): much much more space, never feels crowded, windows all around with views on the airport and sometimes the Alps, a nice terrace with chilling area, a la carte and buffet dining area , huge bathrooms some with bathtubs, great service. Seriously, unless you're a gourmet buff, this is better than CDG (and this is not to say that the food is bad, in fact there is more choice and it's top quality. It's just that the dishes aren't as elaborate/refined). And I never really like the boxed-in feeling of the CDG Premiere lounge, it has few windows with a view on basically nothing, and most of the lounge is far from the windows.
• ZRH: flight-wise I find this the best. On the 777-300ER, seats have great privacy without being claustrophobic, huge screens, most comfortable bedding. Best catering, in terms of choice, quality and preparation (many staff have been to hotel schools in Switzerland, it shows). Probably highest end amenities (Zimmerli PJs and La Prairie amenity kit, which Lufthansa now also has). Connectivity on all flights. Super attentionate service. Like a 5 star Swiss hotel.In ZRH, if you continue to short haul flight or terminate in Zurich, car service from the jetty to the First Class Lounge just like with AF at CDG, with private passport control, no security (just like FRA and MUC). From the FCL, super short walk to connecting flight if docked at the terminal, or limo service to outside position. If continuing on longhaul flight, there is a First Class lounge right in the concourse where you arrive. No security controls, no passport, just a very short walk. The FCL for longhaul flights (E gates) is a very very nice lounge, very spacious, windows all around with view on airport and spectacular view on Alps, a la carte dining area, good bathrooms. Food not as elaborate as in CDG, but good quality. The FCL in A for short haul flights feels more cramped.

So, all in all, I wouldn't make it such an obvious choice for Air France. It's absolutely great, but not an obvious winner along all dimensions. And the FRA experience is not the best you can get in the LH group for FC flgihts from LAX.

In the end, you can't go wrong with either. Let us know what you'll do. And maybe let us know to where you're connecting to.