Open System Preferences > Flash Player then select the Advanced tab.
Click Delete All under Browsing Data and Settings
Back to Safari. Press Command + Option + E to empty the Safari cache.
Quit and relaunch Safari, try a video.
If that doesn't help, you may have the YouTube HTML5 beta enabled.
Opt out here > YouTube
Linc, thanks for your help. You solved my problem (although I forgot to mark "solved"). I was fast becoming displeased with Safari, but peace has been restored.