I have the 200-500 and it is an excellent zoom, I like it very much! It is sharp, has great colour rendition, great bokeh for this type of zoom and is very well built. he AF of the 200-500 is not what I would call fast, but mediocre or adequate in good light dropping off in low light.
I purchased it as a lighter weight zoom for traveling when I want to do birding as well as the usual holiday/travel landscape photography etc. My usual birding lens is thew 400 f2.8 to which I add TC's as required which also gets me 560mm f4 with the 1.4x TCIII, 680mm f5 with the 1.7x TCII and to 800mm f5.6 with the 2x TCIII. However, this set up is usually too large and heavy for travel and thus the 200-500 purchase. So, if you are going to use this as a primary birding lens then the Sigma Sport is possibly a slightly better option simply because of the extra reach and it is supposed to be a tad sharper at 500mm as well as 600mm. It is slightly heavier than the 200-500, though, and would need to be factored into any decision. I did try the Sigma Sport in the shop and it is also well made, but noticeably heavier and has a very heavy lens hood. Personally, I just didn't like the feel of the Sigma lens. However, if I was going to go for a lens that would serve me as my primary birding lens, then I would possibly get the Sigma due to the extra reach and the fact that it is supposed to be a tad sharper - relying on reviews for this assumption.
A good review at Photography Life which also has a comparison to the Sigma Sports (and others) on the sharpness front:
https://photographylife.com/reviews/...0mm-f5-6e-vr/3
A good review of the comparison between the other "Super Zooms" here:
https://photographylife.com/nikon-20...ma-150-600mm-c
Just remember that this is at one specific camera to subject distance which doesn't mean that the sharpness is the same at other distances. In other words, these test chart sharpness results are only a guide and do not reflect the particular lens's overall performance.
The 300 f4 PF is also an excellent lens, stupid sharp bare and still extremely sharp with the 1.4x TC attached for 420mm f5.6 and also sharp with the 1.7x TC attached for 510mm f6.7. It is ridiculously light and easy to handle and if you are after lightness and compactness and can forgo the slightly more versatile zoom, then I would recommend the 300 PF and use TC's.
My 200-500 gallery:
http://www.pbase.com/lance_b/nikon_200_to_500
300 f4 PF gallery - this is not my lens but one I borrowed to try out:
http://www.pbase.com/lance_b/nikon_300_f4_vr