While F2.8 is largest opening of the lense, it does not mean it's used at that opening! I only shoot at F8/11, and under few circumstances F5.6. I'll try to answer your question though : There are two openings in a lense, one that we control ourselves (F-stops), and the other is permanently set. My 28-300mm was F4-6.3, which means the lowest aperture I could set was at F4 (at 28mm, and at 300mm F6.3), but even at F8, the light is going through the permanent F4 (or up to 6.3 depending on focal length) setting. Now on F2.8 lense, your "permanent" aperture is F2.8, regardless of what F-stop setting you set. Confused? Read on... here's why they call it "fast-glass". On my 28-300mm, at 200mm it would be roughly F5.6. On the 80-200mm, at 200mm I'm still shooting at F2.8, as the "permanent" aperture. So, say we're shooting at F8 on both lenses. On the 28-300mm, I'd get say 1/250th at F8. On the 80-200mm, it's exactly two stops faster 1/1000th at F8, or 1/500th F11, or even F16 at 1/250th! So, now you can see just how much faster the "glass" is... we have many more options, intead of just F8. When you buy an F2.8, you're buying top-quality optics, so it's fairly heavy (compared to slower F4 or F5.6 lenses). As far as sharpness, the sweet spot on all or most lense anyway is F8 (in some cases F11), and as I said, the glass and quality altogether is much better on more expensive lenses, I guess that goes without saying though.
While writing this up, I may have discovered why vignetting occurs, correct me if I'm wrong. On an F2.8 lense, you should never get vignetting, because the "permanent" F-stop can never be larger than the variable F-stop. Where as on the 28-300mm F4-6.3 (for example) at 300mm, the "permanent" F-stop is F6.3, but the lense can be open to F4. The variable F-stop is closest to the body, where as the "permanent" or constant F-stop is further out. So, imagine making a circle over your eye with your fingers, and another one further away, but much smaller, this is exactly what's happening to the lense... and the result is vignetting! To avoid the problem, you'd have to shoot at F6.3 or smaller (F8, 11, 16) . So, that's why shooting on an F5.6 lense at 200mm, you can't open the lense anymore than F5.6, or you'd get vignetting, but because the lense is slow, you basically have little choice. I hope I explained this properly... it makes sense to me, but I may not have explained it well.
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