Done both (JL ~10 J segments to/from North America, CX ~100??) My 2cents:

CX wins
*"Equality" of seat choices. None of them are bad on 77H, the 77W bird that flies to both BOS and ORD. Cirrus herringbone is excellent. Versus JAL, has a superb hard product but I dislike the aisles. So 4/6 or 4/7 (787 or 77W) seats I find inferior. I actually like the middle on JAL's 77W (only 7 across has this), because it's quite private albeit skinny. Obviously windows are the best on JAL. And you should note that JAL holds off seats for certain fare types regardless of elite status, while CX opens all its seat selections open to certain OW status members regardless if you bought a cheapo fare. I hate this but....it's an advantage in your case.
*IFE. (credit to kaka). Don't be deceived by JAL's huge screen. The interface is clunkier than CX, and if you're an English or Chinese speaker CX is faaaaarrrr better for selection it's not even close.
*Seat overall This might be controversial, but I think overall reverse herringbone is superior to the JAL suite concept thing in J, Apex or whatever it's called. If you snag a window in JAL you're winning, but you still don't have much storage next to/around you. This is related to seat equality, but overall I tip the hat to Cirrus herringbone. This is tough though because JAL's product is still great. I just hate getting the aisles, I've been accidentally bumped while sleeping on more than one occasion. Pretty impossible in CX's Cirrus product, especially when the aisle guards are raised.
*HK lounges by CX beat JAL's lounges in Tokyo.
*Cool cabins I am baffled by the broad stroke "Asian airlines have hot cabins" generalizations I read on the blogs, from presumably (un)frequent fliers who think they are. CX cabin temperatures are generally set to 22 C (71.6 F), although I've seen a mix of 21 (70 F) and 22 on recent flights. While it is true sometimes they're set to 23 C (73.4 F), they're rarely stifling unless it's a genuine mistake or oversight. And they are always amenable to turning it down you just ask. If it's at 24 or 25 that's an accident (or definitely unacceptable), and they will apologize and reduce it. On my JAL flights I didn't see the temperature, I certainly wasn't roasting as I read concerns, but it wasn't cool like CX. My general feeling is the biggest complainers are folks who don't fly long-haul more than a few times a year, and don't realize just how damn dehydrated the 77W makes you after 6-7 hours, which makes your body temperature feel significantly hotter than it actually is. The plane is big and powerful but it is not the world's most comfortable to fly in. Anyway, I like it cool but usually don't have any problems on CX. And each time I think I have a problem and check the temperature, it's indeed set to 21 or so, which means the problem is me.


JAL wins
*Catering. Not even close. JAL has first meal off the cart, and rest is a la carte. Better wine, better food, all around better experience period end of story. To be fair, I'm a regular on CX so I might be overly-critical, but I don't think any objective observer will come to the conclusion CX J is better in this regard even if they don't fly either often. And granted, CX has some decent mid-flight snacks like the burger and noodle soups. But JAL wins clearly in my book. Especially so if you like wine, CX's J wine is nearly un-drinkable until they do the fleetwide upgrade with the new service wines. Not sure when this is going to happen.
*Wifi. CX doesn't have it on the 77W and JAL does so.......JAL's isn't the world's best in-flight WiFi, but CX is a 0 here.
*Peace and Quiet I don't have a huge sample size, but never had a big disturbance on JAL J class. The peace and quiet attitude is similar for Japanese lounges, trains, and just about everything else. Voices low, respect and quiet everywhere. On CX, I've had "regular" disturbances you find in the rest of the non-Japan world...kids running up and down aisles, passengers holding loud conversations, flight attendants getting chatty in the galley, etc. It's not totally normal on CX, but I think you have higher odds of a disturbance on CX than JAL. I hate it when this happens on CX, especially if I'm trying to sleep. This is mainly a Japanese culture thing moreso than JAL but still a win for JAL in my book.

Wash
*Bedding. JAL has this small mattress pad thing which CX doesn't offer, but CX has a much superior duvet to JAL. JAL's pillow is better though. A lot of talk is made of mattress pads but I find CX's seat cushioning to be very well padded and comfy, for me the difference maker is the duvet. CX has one of the best business class duvets of any airline.
Service. I think this comes down to personal style. I like CX (I speak English and Chinese), but I can imagine if you spoke Japanese perhaps you like JAL better. I think if you speak Chinese you really can get just superb service on CX beyond what is discussed in these boards.

I'd probably just go based on schedule, price and what your US connections are like.