so adobe uses stub installers, which means that it is a small program that goes out and gets the program. The reason for this is because of the download page; that page contains an optional McAfee installer, that is the reason they use the stubs this way it allows them to bundle that extra bloatware.
There are a couple options for you to do:
1) Preform a fully manual uninstall and re-install kind of a pain.
2) A clean install if things do not seem to work, even more of a pain.
3) Use a scripted uninstall, clean up, and re-install
To do the uninstall you need to get the uninstall file here:
Right click, and run as administrator
Right Click, and Run as administrator
If these files are not running for some reason, try a reboot and run them, if they are still not running then you could have something wrong with your system, you Antivirus preventing it, or a piece of malware or virus.
IF the manual install does not work try going here: Adobe Flash Updater - Browse /BAT version at SourceForge.net
get the downloadFlash_XPandUP.cmd file. ( you can right click and edit to see contents. Basically this file will do the uninstall, and clean install of your plash player versions for you. Once again right click and run as administrator.
Judging by the fact that you were able to install the older flash player; which is the actual reall install not the stub seems to indicate something is blocking the stub programs to run.
Hopefully that gives you some direction
This may happen because of a certificate issue. The installer uses the SHA256 certificate, which may not be compatible with a computer that is running Windows Vista SP2.
If this is your setup, head over to the Microsoft Support page to download an update for your operating system.
Thank you Maria. I do believe I am using SP2, so that may be why it won't install.
I was able to update FP16 to FP20 by using the Adobe Flash Updater - Browse /BAT version at SourceForge.net
as suggested by Carm01.