Reboot your machine!
The new version shows in IE straight away. Strangely, if you check your installed add-ons in Firefox immediately after installing a new version, it shows both the old and new versions. A reboot removes the old version, leaving the newest one to take precedence!
Exactly the same thing happened to me. It even showed the latest version in 'uninstall programs', so I knew it was properly installed and a reboot was probably required.
Unbelievable! Even after 30+ years using computers, and having once been a Contributing Editor of Personal Computing magazine, it never even occurred to me to reboot the PC!
Although, who would ever imagine that a browser plug-in would require a system-wide reboot! Indeed, with Opera and IE showing the new Flash version -- and with prior Flash updates showing the correct version right away on Firefox -- it never crossed my mind to reboot today.
Also, hey Adobe, why not advise users, when they install Flash, that they will need to do a reboot before Firefox uses and/or shows the correct new version!
Thank you, Steve, for solving this mystery and taking the time to post the answer!