The price to pay for half a stop more speed over the 300/4 + TC17...
I would expect a number of advantages.
With TCs the autofocus gets progressively worse as the TC power increases. I would expect significantly better AF from a prime 500mm f/5.6 than a 300/4 PF with TCs or the 200-500/5.6.
The 200-500 is quite good from 200mm to 400mm but in the outer areas of the frame (outside DX area) the 300mm PF is clearly sharper than the 200-500 at 300mm. The zoom loses some sharpness going to 500mm (compared to shorter focal lengths) whereas I would expect comparable sharpness from the 500/5.6 PF wide open as we get from the 300mm PF. Longitudinal chromatic aberration is likely to be well corrected.
Because there is only one Nikon PF lens so far, guessing what the 500 PF will weigh has to have wide margins of error. A conventional ED 500/4 is about 3.9kg, halving that weight to get the smaller maximum aperture makes us at 2 kg but then there is the PF factor which can reduce weight by 30%. I think maybe 1.4kg would be realistic. This would make it substantially lighter and more pleasant to handle than the 200-500/5.6 at 2.3kg.
The 300 PF images can look a bit drab on cloudy, dim days and in such circumstances I prefer conventional refractive optics. I am the most pleased with the 300 PF's handling of bright conditions and coloured, textured subjects. However, in soft low light I would like to have a 300/2.8. Even though the 300 PF has shortcomings I have taken shots I could not have with heavier lenses, just because it's so easy to walk around and shoot with the 300 PF and the autofocus is quite fast and accurate (with no TC). I would expect the 500 PF to be similarly fun to use but the AF may be a bit less good than with the 300. Multi-CAM 20k supports 45 cross points with f/5.6 lenses instead of 99 one gets when using the 300/4 PF. Of course an f/5.6 lens lets in less light so the ISO will have to go up in many circumstances. I guess it is a bright outdoor light lens mostly, where it comes to photographs of moving subjects at least. But I would expect that people would enjoy shooting with a travel friendly high quality long lens for various reasons.