I have seen both this and this, and I am asking "the other side of the coin", in a way. What I was wondering is: why didn't we make the play on the Think Different campaign in the URL as well as in the site? It is strange to have a site called AskDifferent whose URL is apple.stackexchange.com… Why not askdifferent.com? After all, SO has stackoverflow.com, not .stackexchange.com, and same goes for AskUbuntu's askubuntu.com…
I'll hazard some thought as I was around when this was discussed with the moderators at that moment in time. I have speculation and opinions, but they are just that other than links to other posts.
The people that run the site registered http://askdifferent.com and point it to the name apple.stackexchange.com and the why is for them to answer. I think it had to do with not setting more precedent in making a custom name other than the original trilogy or a preference to market/brand things consistently for web market share numbers or appearances in building the reputation of the network. Having a custom domain necessarily weakens the link to the branding of the company and some process or some one made that call a while back.
The community has voted pretty overwhelmingly to have things the other way around - have the site served out of a custom name and have requests from the apple.stackexchange.com resolve to the custom name, but those are just internet votes and not anyone that has actual authority to make any change.
This is covered (for all SE sites) in this blog post.
I believe SO, SU, AskUbuntu, and any others have those names purely for historical reasons. SO because it's SO; I think AU was independent before it joined SE; not sure about SU, maybe just because it's old enough to have slipped through.