First of all congrats for your first 600 F4! The AF-S II is pretty capable lens and is also the lightest of the "old" 600 F4 lineup (guess up to the FL version). Unfortunately like all the older designs is quite front heavy, so using it handheld is very difficult and tiring, it better belongs to a sturdy gimbal.
I had it for some time (in the pretty rare and fancier light grey version, pretty amusing to go around and shocking fellow photogs when they got closer and discovered it was not a Canon L lens :P) and quite like it, AF was fast and image quality was very good, although wide open definitely not up to par with the best long lenses I had the luck to try. Like a lot of the pre-nanocoating Nikon lenses, I found the shots at F4 lacking a bit of contrast out of camera, with a slight hint of glare when shot in bright light, although the lens is quite capable to resolve fine details, as shown in your shots, so with a bit of post processing most of the shots can then be brought to life with excellent results.
I then found a pretty good deal on a good sample of AF-S 400 F2.8 G VR and decided to go for it, but I managed to have both for a couple of weeks before at the end decide to sell the 600 F4 (even if begrudgingly, but couldn't justify to myself to keep both as a simple amateur).
Here my observations based on using both lenses on a D810 and a D500:
- AF of the naked lens: the 400 is faster, not by much but it has a slight edge
- AF of the 400 1.4x TC III vs 600 naked: here the 600 had the edge, but not by a significant margin, just a tad faster
- Image quality: at least my copy of the 400 is definitely sharper than the 600, even wide open and by quite a good margin I'd say. The thing that finally sealed the deal for me in this respect was that the 400+1,4 TC wide open (i.e. F4) was sharper than the 600 naked at F4
- Contrast: the Nanocoating on the 400 helps quite a bit with the perception of microcontrast out of camera, especially when shooting in bright light. I like that more "moderrn" look more, but someone could also prefer the more "laid back" drawing style of the 600 D II (not me)
- Handholdability: the 400 weights basically the same as the 600, so it's not exactly handholdable, the only catch being that it is a tad shorter so it's a bit easier to balance it against the body for the occasional handheld shots, not to mention that VR helps quite a bit to get sharp shots in these occasions (of course it remains a lens that has to be supported for the best results).
- Flexibility: paired with my D810, the 400 F2.8 is, ahem, a 400 F2.8 lens, so I can use it also in dimmer light conditions or for something else that is not nature photography
- TC compatibility: the 600 has a strange bug, no matter how I tried, and even if on paper they should be compatible, I didn't manage to make the AF work when I tried it with the paired TC 1.4x II, so you may want to check the web for more info, in case you aim to buy one. The 400 worked flawlessly with both II and III versions.
Having said that, if not for the 400, I'd have kept the 600 and be a happy camper nonetheless, it's still a very good "big gun" and it has a butter smooth bokeh, so enjoy it to the fullest! 
