I have both and I use both. The 1DX is a better camera overall, but the 5DIII is also a very good camera. It is smaller, lighter, it has a better quiet mode and it has HDR. The 1DX equals or betters everything else. I have the battery grip for the 5DIII, but because I then loose the weight and size advantage, I never use it.
If you avoid fast action and very high ISO conditions, the performance differences are behind comma in my view. I often hand hold a 600mm f4L IS II, often with the 1.4xIII extender. Over time that is a lot easier with the 5DIII. But when AF conditions become tricky and I need more fps, nothing beats the 1DX.