I just noticed that Stack Overflow employs Firefox only CSS to round the corners of the user badges on the front page. It's an interesting idea but what would be the pitfalls and advantages of using Firefox only CSS (aside from the blatantly obvious ones)?
-moz-border-radius:6px;
-webkit-border-radius:6px;
Above: The CSS used to round corners on the Stack Overflow front page.
Pitfalls are of course that it's not supported in other browsers and its not in the W3C specification.
The current implementation is bad (both in firefox and webkit) since they do not share the syntax.
The advantages are that it's much easier than using corner images or other tricks. The obvious disadvantage is that your page doesn't render as intended on IE, which is still used by more than half the world's web users.
Overall it's situational; I imagine SO gets a higher than usual proportion of people using Firefox, so the IE issue is not as relevant. And rounded corners are a minor visual improvement, so if some people see it and some don't, it's not that big a deal.
It works in Firefox and the Webkit-based browsers (notably Safari and Chrome). There are no alternatives for IE or Opera.
The main reason to use it is when you want to give rounded corners to elements that display on top of patterned or unpredictable backgrounds, which is not possible otherwise. Normal CSS and images can take care of other situations.
It's also ridiculously easy to implement and surely works for more than half the people on this site.
The main problem would be, as I see it, that your css wouldn't validate. Other than that I see no reason why not to use this type of platform specific features as they don't cause any harm to users whose browsers don't support the features.
I agree with BeefTurkey.
I might even go further and call it a case of Progressive Enhancement with CSS. To borrow liberally from Understanding Progressive Enhancement, I'd consider rounded corners to be part of the colorful candy coating around the chocolate-covered peanut.
And eventually CSS3 will be ratified and border-radius will be standardized. -moz, -webkit and -ms prefixed styles can be removed and replaced with their standard equivalents. Your CSS will validate and people using browsers that don't support CSS3 will still have a completely acceptable experience with people using browsers that do support CSS3 will get an enhanced experience.
Alternatively you could continue to use the proprietary CSS in addition to any standards to give an enhanced experience to an even larger audience. It would really depend on how much effort maintaining all that CSS requires and how important you feel the enhanced experience is to your audience.