There is a blog post from Joel on the Stack Exchange blog around the time of WebApps' Area 51 Graduation:
A while ago, I wrote:
“Individually-branded sites felt more authentic and trustworthy. We thought that letting every Stack Exchange site have its own domain name, visual identity, logo, and brand would help the community feel more coherent. After all, nobody wants to say that they live in Housing Block 2938TC.”
Well, funny thing… that didn’t quite work out the way I expected... mostly because nobody could think of any good domain names. Believe it or not, “NothingToInstall” was one of the better suggestions. Ack.
This reasoning was used as justification for reverting "Nothing to Install" to "Web Applications", and maintaining graduating sites as "topic.stackexchange.com".
I understand the backlash that Nothing to Install received, and I understand that some sites' names are just difficult, wordy, mildly ambiguous, and aren't the obvious solution to the problem. But let's flip this situation on it's head.
Ask Ubuntu has graduated and maintains it's own domain identity, that being askubuntu.com, which ubuntu.stackexchange.com redirects too. I'm well aware that this is in large part to the partnership between Stack Exchange and Canonical. Here in pure Stack Exchange land, though, there are a handful of sites that actually have and actively maintain an identity (Ask Different, Seasoned Advice, Cross Validated), spectacular names, and the domain names to match.
Then there are those that were fraught with contention and didn't have an overwhelming agreement regarding the name/identity of the site. Nothing to Install -> Web Applications, and personally-notable, Gaming.
Everyone is correct here. Domain names are important, and both topic.stackexchange.com and an actual identity serve a significant purpose. An elevator pitch is important not only for introducing the users to the site upon their first visit, but also to explain what the site is in, well, elevators. All of these work together.
However, there are sites that have developed and earned their identity. I admit to speaking with absolute bias with regard to Ask Different, but also am incredibly fond of Seasoned Advice. To date, Ask Different has accepted users publicly for over 1 year now, and has benefitted tremendously from external advertising, the WWDC and Lion release traffic, and stands to continue this trend from iOS 5, and the continual release and update of hardware by Apple. There has been one question about the name of the site, but this is far from the mixed opinion that other sites have encountered.
In general, neither approach is correct in all circumstances. Forcing every single site to not have their own identity or branding, but to simply be subsumed into the generic "stackexchange.com" domain is, in some ways, just as harmful has forcing all sites to choose their own name, domain, and branding.