Another difference is that, most of the time, fast lenses (say f/2.8 instead of f/5.6) are better quality lenses. There is no law of optics making this so, it's economics. Manufacturers rarely spend whatever it takes to make a really high-quality lens in (e.g) 50mm f/3.5 (which is slow for a 50mm) or 400mm f/6.3, and when they pull out all the stops to make a fast lens (e.g., 50mm f/1.2 or 400mm f/2.8) they also take the trouble to get just about everything else as good as possible.
The result, in practice, is that a fast lens is usually a good quality one, even though you may not be actually using the maximum aperture. It is also nearly always quite a bit bigger, heavier, and much more expensive.
Finally, only a handful of the very best lenses perform at their best wide open. Most lenses are sharper a stop or two down from their maximum aperture. So, if you want to shoot at f/1.8 with a 50mm lens, you could use a typical cheap plastic f/1.8 lens wide open (which will return a so-so result) or an expensive f/1.2 stopped down to f/1.8, which will not only provide you with better quality glass in the first place, it will also be performing close to its best because you have stopped it down a little.
With all that said, don't worry about it too much. The real difference between top-quality glass and cheaper lenses isn't really image quality in good, normal conditions. Let's face it, depending on the type of photography you do, most of the time you will be shooting at f/8 or f/11 anyway (for depth of field) and at f/8 or f/11, nearly all lenses perform pretty well.
For example, in good light at f/8 or f/11, there is practically no quality difference between my $130 plastic Canon 50mm f/1.8 II and my $800 Canon EF-S 60mm macro. Similarly, given good light and a modest aperture, at 35mm my old $100 EF-S 18-55 delivered nearly the same quality as my vastly bigger and heavier $1700 24-105.
So why spend the extra? Obviously, a cheap, slow lens can't do the low-light or shallow depth of field shots that an expensive fast one can do. Sometimes this matters, sometimes it doesn't: depends on the sort of photography you do. More importantly, in my view, top-quality lenses (which are usually fast lenses) seem to have a way of performing well even when the light is bad. I'm not sure why this is, but I suspect that it is probably nothing more than the accumulation of a lot of small differences, the sum total of all those small, expensive, attention-to-detail tweaks the manufacturer made when putting the expensive lens together.
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