How to fix Admin Menu that's broken on live site, working on identical dev site
It was a file permission problem! 

It turns out that, as per Apache's and Drupal's recommendations, I had set the permissions for all folders/directorys in my sites/all/* folder to be 750, and 640 for files. So for those who don't know, that means that 'others' (people who don't 'own' the files or folders) can't access them in any way.

It seems that my web host had incorrectly not assigned Apache to the correct group so that it would be able to run using those safe permissions (I know why, and I'm going to send them an irritated email asap).

I simply used Filezilla to recursively change the permissions to 755 and 644 respectively. Not quite safe, but I can change them back later.