Since version 18 Firefox now disables older versions of the Java plugin and the user has to click on the applet or the red brick icon in the address bar, then click Activate to make the plugin load. This is a good thing, but it can be annoying. Even though you may have the latest version of Java installed there may still be an old Java plugin lying around making your browser insecure and causing Firefox to block Java from loading every time.
I found this excellent article on Mikes Musings regarding the Firefox plugin loading mechanism: http://mike.kaply.com/2012/02/16/how-firefox-loads-plugins/
Using that information I discovered that JavaFX contained an old NPAPI plugin which was causing this.
These are the registry paths I found that Firefox uses to determine what plugins to load:
After uninstalling JavaFX Firefox loads the Java applet without prompting.