Some Youtube videos use HTML5 whereas others still use the older Adobe Flash Player. Perhaps HTML5 is working okay on your computer but Flash Player isn't? (It's notoriously fickle).
It won't do any harm to reinstall Flash, so it might be worth a try:
First, go to your list of installed programs and remove any references to 'Adobe Flash Player'. Then RESTART YOUR COMPUTER. (That's IMPORTANT!).
Next go here:
https://get.adobe.com/flashplayer/
Deselect the 'optional offers'.
Click 'Install Now'.
Once the file has downloaded (in a second or two), click the downward-pointing arrow at the top left of your screen and then double-click on the installation file to run it.