If there are problems with updating or with the permissions then best is to download the full version and trash the currently installed version to do a clean install of the new version.

Download a new copy of the Firefox application and save the disk image file to the desktop

• Firefox 24.0: http://www.mozilla.org/en-US/firefox/all.html

• Trash the current Firefox application (open the Applications folder in the Finder and drag the Firefox application to the Trash) to do a clean (re-)install
• Install the new version that you have downloaded
• https://support.mozilla.org/kb/Installing+Firefox+on+Mac

Your personal data is stored elsewhere in the Firefox profile folder, so you won't lose your bookmarks and other personal data when you uninstall and (re)install Firefox.

• http://kb.mozillazine.org/Profile_folder_-_Firefox

Try the following:

1. Go to about:config in Firefox
2. Locate the string app.update.channel
3. Ensure that the value of this string is release

Please report back soon.

You can manually install the most recent version of Firefox from the Mozilla website if you wish to update as soon as you can.