Why would I choose Nikon 17-35mm over the Nikon 16-35mm, or vice-versa?

The Nikon 17-35mm is obviously more pricey than the 16-35mm. However, with the new 16-35, you get a dust and weather resistant lens that is capable of braving the rigid landscape works. Weight is also a factor since the 16-35mm won't strain your shoulders unlike the hernia-inducing 17-35mm. Based on other reviews, the 16-35mm is an amazing ultra-wide lens almost if not at par with the Nikon 14-24mm when it comes to sharpness. At this point, I will be needing your honest to goodness answers since I plan to take any of these lenses for travel, landscape, and all-around, go-to lens? I'm also considering the Nikon 24-70mm, though.

Added (1). Is it just the maximum aperture of f/2.8 vs.f/4?

If you do not see the advantages of a lens that has a maximum aperture of f/2.8 over one with f/4, you may not be ready for either of them, not until you learn how to use the camera and lens you have already.

It you don't need, want, or can't appreciate how the f2.8 maximum aperature will give you great images in low light, allow you to capture action with faster shutterspeeds, allow you to use lower ISO's for sharper images, or reveal creamy bokeh in portraits… Then get the Nikon 16-34mm.

In my portrait and sports photography work, a fast optic is a must.

Faster lens have many advantages. The extra weight can be attributed to the extra glass, and the housings that surround them.

As for the price difference, just price a Nikon f/1.8 and Nikon f/1.4 in either the 50mm or 85mm range and you'll see that same huge price gap.

I prefer f2.8 over F4 with VR, who need VR on a 16-35? Another way to increase the price for lens, in my opinion it's pretty useless on a 16-35, but who knows it might save the day.

I don't know were you got the thing about being almost pair with the 14-24, it's far behind the 14-24 optical quality.

The 17-35 give slightly better result than the 16-35 on both aps-c and full frame.

Both the 17-35 & 16-35 have excellent build quality.

Both suffer from soft border and soft corner, the centre is very sharp though.

If you don't need the 14mm and the outstanding sharpness of the 14-24 or the VR from the 16-35, the 17-35mm 2.8 should be a good choice.

If you're concern about the weight, there's 65g difference between these two, it's still 155g less than the 24-70.

If you have a full frame the 24-70 could be a great all around lens, landscape, portrait, street shot, on aps-c it's still a good lens, but for landscape it will be less wider with the 1.5 conversion.

I don't think there's a bad choice in picking one or the others, it just come down to your personal need.

The 17-35 is a D1-era lens which predates the 14-24 by a few years. The 16-35 is önly" f4 but it offers that lovely nano-crystal coating. With the 17-35 you pay a premium for the f2.8 which is only a stop more.
If your main purpose is landscape you don't need f2.8 and then the 16-35 would make a lot more sense.