Which do you use and why? Does one provide any clear advantages over the other?
Admin menu provides a quicker way to get to many of the core admin functions rather than going via admin. It is part of D7 but in D6 is has been known to be buggy and slow.
I use admin_menu for administrators and developers who need quick access to administration functions, but I use admin for everyone else.
Admin's main weakness for administrators, that it only shows you one section of the menu at a time, is also its main strength for content creators and maintainers who would be overwhelmed by all menu options. The fact that you can have a sidebar with just Create content is great for data entry.
A big plus of admin is its ability to be tucked away into one corner instead of taking up the entire top of the website. In my user testing, people don't seem to think they're editing the live site when the site has a toolbar attached to it: before switching to admin, I would frequently get questions asking how they could see the live site when logged in.
I find them both slower than just using the default system, they both add to page load times and it's very easy to get to the admin page you want using Firefox's url suggestion feature.