I have both. I first bought the 800PF and only a few months later the 600TC. I did not sell the 800PF after I got the 600TC (which, at the moment is almost 2.500 euros cheaper then when I bought it but who could have known that.)
The 600TC is hands-down the best bird lens ever made, thanks to the built in TC. It's optically as perfect as a lens can be at 600 and one of Nikon's best lenses ever made (rivalled maybe by the 400TC which I have never used and the 135 Plena which I own and I absolutely love.)
At 840 it is still crazy sharp and even with an extra external 1.4x, it delivers good results (if shutter speed is fast enough and there is no heat haze.)
If you get only one and weight, size and money are not objection, get the 600TC.
The 800PF is also a very sharp lens. Obviously it is less flexible than the 600TC and "only" 6.3 instead of 5.6 (and 40mm shorter).
But: the 800PF is less bulky, a bit shorter and almost 1kg lighter. That is great when carrying the lens. The smaller size can also be important when putting other stuff into your backpack. The 600TC takes up a lot of space. That's why I've kept the 800PF and I still use it regularly.
(I also have the 600PF and whenever the situations allows it, I actually prefer this one, because it is so small and light but often I need f4 or 800/840 mm).
I find both PF lenses to be very sharp, already wide open, and I got some good results when I used them with the 1.4x. I do not, at the moment, own the 2x.
So, from my point of view: If you can get both, maybe this is the best solution, otherwise go with the 600TC if you can live with the drawbacks, you won't be disappointed, it really is that good. If you rarely shoot at 600mm because it's too short and don't mind f6.3, the 800PF is a great choice. And for the money saved, you can get a few other lenses or an extra body or go on a long trip somewhere.
In the end every lens is a compromise. So far, I am super happy with both these lenses.
(I shoot both lenses with two Z8 bodies, I have never used another body with these lenses.)