The Firefox 15 version that you use is very old and very insecure with a lot of disclosed security vulnerabilities.

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 29.0: 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

Download the full installer and run it.

http://www.mozilla.org/en-US/firefox/new/

Can you explain in more detail what would be the problem?

It is not clear to me what is causing this issue, but we've seen this often reported when users with an old Firefox version try to update. If can be caused by Firefox using a link to check for updates that is no longer supported on the Mozilla server and that causes Firefox to hang because the server isn't responding in the way Firefox expects. I assume that the connection will time out sooner or later, but nothing will happen.

It is best to download the full version of the current release in such a case.
That also prevents having to download and install multiple intermediate Firefox versions before getting the current release.