As of the 1.02 patch, the game should automatically detect that it's running on the Pro and enable higher-resolution super-sampling, improved dynamic lighting in the cockpit and higher-resolution screenspace reflections.
Of those three, the dynamic lighting in the cockpit is the most obvious - sitting near explosions and other light sources should light up the cockpit in a more realistic manner than the 'standard' version of the game.
The increased super-sampling resolution helps to reduce shimmer by increasing the amount of anti-aliasing - you should find the cockpit text is easier to read on the pro, for example. Ultimately the final display resolution is limited by the headset hardware so it's not going to be as obvious as, say, going from 1080p to 4k on a standard "2D" game.
Higher-resolution reflections means there's less 'popping' of elements in the reflective surfaces in the game (like the shiny floor areas in some levels - glowing lights reflected in these can often flicker or pop with the lower resolution).