You can see more details (blood/dirt) without 4k monitor. However, 4k monitor is needed to see it at it's full capacity.

The reason it looks so different is, as I wrote before, the Depth of Field effect in play on the Pro, which DigitalFoundry itself notes in its description of the comparison. Yes, the X version is going to look better, but that effect makes the Pro and PC versions seem far softer in comparison. The X version will look blurrier, as well, if Nixxes institutes the same effect, and the Pro would look quite a bit sharper with it removed. The game will still look better than the Pro version, as it has a native resolution and higher texture quality, but this screenshot is highly misleading. 

Here is another image.
The game may as well be an xbox one x console exclusive considering how good looking it is.

The One X is in native 4K in the video and the PS4 Pro checkerboard 4K.

What Resolution is the ps4 pro checkerboarding from?

1800p

Wow! the level of detail in the face on the x1x is super compare to the pro. -- ain't no denying this looks better on x1x for sure.

Read my other two posts, please. Yes, the X is going to look better, but the Depth of FIeld effect that the Pro is using is the main reason for the perceived difference in quality here. The actual differences are more subtle, as in the X version has sharper clarity and better textures. 

What is the difference is between native 4k vs 2160p checkerboard?

The thing we have to remember is this difference is going to apply to many other games too. Pretty much all multiplatform games should be consider xbox one x console exclusive if they have these big differences.