I have been fortunate to be able to use all of Nikon's current offerings, so what follows is my subjective impressions, based on my experience of these with the Nikon 70-200mm f/2.8 VR and 300mm f/2.8 VR (mark one versions both).
First, up the TC-14e II. This is easy: you lose one stop and gain 40% in focal length. That's it. In my experience, with the aforementioned lenses you lose nothing else - no significant drop in autofocus performance and no significant drop in image quality, even wide open. Bearing in mind that makes my 300mm a 630mm f/4 on a DX camera, that's a fantastic result. Also, the lightweight, portable, hand-holdable nature of this TC on the 70-200 on a light camera like the D7000 is a great shoot-all-day combo when lugging a bigger lens around is impractical.
So, since teleconvertors are all about adding length, what's the TC-17e II like? Well, perhaps surprisingly, it performs just as well as the TC-14e, except instead of 40%, you're getting 70% more focal length (so 300mm becomes 510mm on FX, 765mm on DX!). Yes, you lose another half-stop, but even shot wide open (f/4.8) on my 300mm, the results are stunningly good. I don't even detect much slow-down in AF performance, at least not on a D3 series body. Again, no problems on the 70-200mm, just a very portable solution that delivers great results.
In terms of adding length, the TC-20e III is obviously the daddy. This latest version of the 2x TC has an aspherical element and is supposed to deliver exceptional image quality. First of all, my disclaimer: I never use this lens on my 70-200mm. Why? Well, if I really need that extra length I just switch to the 300mm and the 1.4x. In my opinion, if you're even contemplating buying a 2x convertor to extend a zoom you are barking up the entirely wrong tree. Get the longer lens you really need or get closer to your subject (buy a hide, improve your fieldcraft, whatever), or just go with the 1.4x or 1.7x.
On the 300mm, at first, the TC-20e III delivered disappointing results. After the other two, I naively expect to click it on and go, but it's taken a while to get to grips with it and get it to deliver results near to the other TC's (note, I'm saying near, not identical to). I discovered quickly that I needed to close down 1.5 to 2 stops from wide open to avoid disappointment with the image quality.
Not a huge problem perhaps, but we're now down to f/11 on the 300mm, and that two stops has an impact on shutter speed that is perhaps related to another issue that I have found. Previously I would leave VR switched on whether the lens was mounted on a tripod or not, as per the manual, but with this TC the image quality seems noticeably better with VR off. Since discovering this, I routinely disable VR except when handholding, whether I am using a TC or not.
So, with VR off and stopped down the TC-20e III does deliver reasonably good image quality on the 300mm f/2.8 (see previous post for some examples). That loss of 2 stops is a problem when photographing small garden birds in the UK though - they move so quickly we really need to keep those shutter speeds short, so inevitably it involves raising the ISO, potentially degrading the image quality further. But, bearing in mind that these birds are so small, this compromise seems reasonable to achieve an effective 900mm lens (DX), especially when you consider the price of super telephoto lenses...
To sum it up, I could only recommend the TC-20e III if you were shooting DX (I don't see the point in adding this on a FX camera), and you really needed the longer length for specific subjects but couldn't justify the cost of a longer lens (which is fair enough as these days, new, they cost as much as a small car).
The TC-14e II & TC-17e II are harder to separate, but if you can only choose one, then the only thing that I can see to separate them is half a stop. Given that the whole point of a TC is to give you extra reach, I would wholeheartedly recommend the 1.7x, and right now it's actually marginally cheaper than the 1.4x in the UK.