Some designers are very, very precise. As they are making detailed decisions about placement and graphics at a pixel level, they don't want to see their work appearing differently than they had envisioned.
Of course, as a CSS/HTML author I know that you can't guarantee 'perfect' placement of items, precise text sizes, etc. across browsers and devices.
I tried to persuade the designer I work with to view things in a more lenient way. On the other hand, the way she has pushed me to implement her ideas of perfection has encouraged me to be much more meticulous and aware of advanced aspects of CSS that I didn't think about previously - letter spacing, line height, different styles of positioning within containers, and so on.
Believe me, for some projects, pixel perfection IS possible across at least Safari, IE, Firefox and Chrome, because I've sat there for hours (days?) on some projects with the Pixel Perfect plug in, the browsers and Gimp making sure that everything matches.
In summary, striving for pixel perfection ensures a professional design comes across as the designer intended, and encourages the implementer to be aware of every last detail.
As for why they give more attention to the visuals than the usability and accessibility of a site, it's hard to give a concrete answer. I'm sure different clients have different reasons. Many designers come from a Photoshop/Illustrator graphics and print type of background, so while they have spent scads of time learning to manipulate pixels, they haven't spent as much time thinking about how websites work, and how people use them. Visual perfection is a concrete thing, so it's either right on or wrong to them. It's easy to tell your programmers to fix placement, not so easy to figure out in what way to change behavior. However, these people clearly don't understand (and I can sense your frustration with this in the question) that CSS and HTML are meant to be flexible, and certain decisions are left up to the browser by design.