From LAX JL flies their newest SS7 product, which is staggered and allows for aisle access for all seats. Many also prefer the food on JL to CX (especially for Japanese and Western food), but out of LAX it will probably be a wash. CX has reverse herringbone seats on most (all?) of their long-haul flights, including to LAX.

CX will get you closer to your ultimate destination if you're traveling to BKK or TPE; JL is better for flying to ICN. With JL you're very likely to have a regional configuration, while CX will have a mix of both long- and short-haul aircraft.

If you've tried CX, and there is availability in JL J, I would try JL so you can compare them both.