I own both cameras, so this is my take.
1. Weight, size and frames per second are the big differences between them. The difference in mp is notable as well, but I consider that to be secondary. If you do a lot of hiking for landscapes, the smaller size and weight is a big factor in favor of the 5D Mk III. Frames per second aren't usually a factor in landscapes, unless you're shooting waves or other moving things.
2. At low ISO settings, image quality is comparable, both are excellent. That said, I've always considered the 1DX to have a "buttery" look to them, really smooth. At higher ISO settings, the 1DX beats the 5D Mk III hands down.
3. The 1DX is superb at bird or other moving object photography. It just can't be beat. The 5D Mk III is also excellent but the 12 vs 6 fps and the buffer size makes a big difference. They have similar autofocus, but the 1DX is better due to its dedicated processor. I've captured many great birds in flight shots with the 5D Mk III that I would not have been able to take with the 5D Mk II, which I also owned.
Most of my shooting is for the birds, secondary is landscape. When I go for landscapes, the 1DX stays home. When I go for birds, the 1DX is on the 600 mm lens and the 5D Mk III is on the 200 - 400 mm or a 70 - 200 with a 1.4X. Both get used and produce great images
Bottom line recommendation, buy the 5D Mk III and save the money for another lens or something else that would improve your photography.