I think it safe to suggest that, in most performance-related matters, the 1Dx will marginally to significantly (though never greatly) outperform the 5DIII.
There are, however, actually a few areas in which the 5DIII outperforms the 1Dx, and they're worth keeping in mind.
First and foremost is price, of course. You can buy two 5DIIIs and have money left over for the price of a single 1Dx. And there are a lot of situations where a second body is far more important than the marginal differences between the two models.
Next, the 5DIII has a silent mode that's not quite believable. For settings where you need to be unobtrusive, such as weddings or contemplative sports, the 5DIII's silent shutter alone will win the day over the 1Dx, all the other superiorities of the 1Dx be damned.
In a similar vein, the 5DIII is a lot smaller. You can slap a Shorty McForty on it and it doesn't look a whole lot different to the non-photographer from a high-end P&S -- especially in comparison to a 1Dx with a 35L. And, yet, the smaller combo will give you comparable results in all but a few extreme and not-especially-common situations. (And, yes, yes, I know...there're people here who think f/2 is slow...you're extreme and not especially common. Congratulations!)
At the same time, the 5DIII works great with the grip (though, obviously, not quite as well as the 1Dx with its integrated grip). Point is, it's very versatile, in that it's equally at home at the party and hung off of a Great White; the 1Dx just can't slim down, while the 5DIII bulks up pretty damned well.
Not to be overlooked, the 5DIII has a few more megapickles than the 1Dx. At lower ISOs, that can (marginally) give the image quality edge to the 5DIII, especially if you're cropping.
In general, if you're considering both and there's a serious question in your mind about which to get, then the 5DIII is for you. It's definitely for you if any of the cases where it does better than the 1Dx are particularly important to you.
The 1Dx is the action photography monster powerhouse. If you're shooting for SI, you'd be an idiot to not be shooting with one. And it's better than any other 1-series camera Canon has ever made, so it's the best choice for all those other photographers for whom that sort of thing is built. If you're shooting in a war zone and you need to be able to club a renegade rebel to death with your camera and still be able to use the camera to take pictures of the steaming corpse, the 1Dx is for you. If you're shooting wildlife in the deep jungle and you never know when you'll have to use your camera to chip off some flint to use to start a fire because your matches got washed down the river, the 1Dx is for you.
But, really...if you're not sure that the 1Dx is for you, then the 5DIII almost certainly is.