I have never heard of the manufacturers using different glass or coatings between different lens of the same range even though the maximum aperture is different. They have certainly missed a huge marketing ploy if they do and that seems very unlikely.  The glass and coatings on say  Canon 70-200 F4 L lens which Danny has mentioned and the glass and coatings used for the 70-200 F2.8 L lens are the same as far as I know. The difference in price comes from the difficulty of manufacturing the larger diameter lens to the same quality as a smaller diameter lens. 
Danny has also made a very valid point about weight. It is slightly off topic, but if I were a sports photographer who had to carry it round all day getting action shots I would be taking the 70-200 F4 and not the F2.8.  Many professionals actually do, and vouch the quality is indistinguishable between the two, the only difference is with one your arms and shoulders are going to ache, with the other not so much.

To get back on topic no lens is going to work as well at the extremes of its range as it does at the median. So yes if you need to shoot F4 get a F2.8 and stop it down and you will get a better image marginally. The difference is not going to be noticed by most photographers though as modern lens are very good at what they do in general. The difference as I understand it would be caused by reflections within the tube of the lens so lessening the contrast of the image. With the larger barrel of the larger aperture lens you would have more tolerance when stopping it down a stop.  However I think the main reasons are ones Danny has touched on. Brighter viewfinder image & quicker focus, most shots are not going to be made at the largest apertures on these lenses in my experience. It is much more likely you are going to be stopping down to F8 or thereabouts to get the dof you need and at mid apertures the difference in the lens is going to be indistinguishable for ones of the same quality. 

As a side note to this, the quality will also fall off if you start using a lens at the smallest aperture. This is caused by refraction of the light around the leaves of the aperture opening.  Hence when I shoot a landscape shot I rarely stop it down beyond F22 even though my lens would go much smaller.  This effect is particularly noticeable on compact cameras where F8 is smaller than a pin hole !

In the previous thread mentioned I suggested a F2.8 lens simply because a 2x converter could be used to obtain 400mm. So in that instance to go for the F2.8 lens gives you great versatility. You would have a great lens operating in the 70-200 range on its own, but is is large enough to add a the converter and still give very acceptable results at 400mm. If you add teleconverters to smaller aperture lens you will slow the focus down and eventually it will not work at all beyond F5.6 on most cameras, and yes I do know there are dodges around that if it is only just on the limits and you have a good camera. For wildlife shots you are most likely going to need the 400mm or even a bit more. 

As mentioned above no lens will perform as well at the extremes of its range as it will in the middle, they are all a compromise in one way or another. You can eliminate one of the factors though, and go for a prime lens and this will be optimised for its length of course and won't have to compromise to work acceptably over a whole range of focal lengths. The same will apply however with regards to apertures as applies to the zoom lens.

Perhaps when someone suggests to you to always get the fastest lens you can, you could ask if they carry it around all day? Ask a press photographer at a golf tournament. I reckon a few of the wiser ones will be carrying the F4 not the F2.8 and for one reason only - the weight ! 

And every one said, 'If we only live,
We too will go to sea in a Sieve,---
To the hills of the Chankly Bore!'
Far and few, far and few, Are the lands where the Jumblies live;
Their heads are green, and their hands are blue, And they went to sea in a Sieve.

Edward Lear
http://www.nonsenselit.org/Lear/ns/jumblies.html