Did you double click on the download to run the installer program?
If so and you're still having problems try completely uninstalling Flash using Adobe Flash Player Install Manager, which should be in your Utilities folder. Then re-download the installer from adobe.com/flashplayer and run the installer again.
Today's Flash Player saga:
Woke up to the prompt to upgrade Flash Player 12 or I'll be blocked from videos. I was supposed to watch something for a training I'm doing.
Installed Flash Player 12. Now I was blocked completely. Checked the Adobe install info.
Made sure my Safari preferences were set to General: open safe files AND Security: allow plug-ins. Yup, already had that.
Called Adobe support Australia. No help there. I'll have to pay $59 for support. You mean, Adobe gave me a gift and now I have to pay them to hand me the scissors to cut the ribbon? I don't think so. That's so wrong on every level.
I uninstalled all Flash Players (11 and 12). Reinstalled 12. Now my computer didn't even recognize that I HAD Flash Player and kept telling me to install it.
Guess what you don't get after install? A "restart" prompt. I restarted my computer and now it works.
Good luck. I hope that scenario works for you.