I recently sent out an update on my personal Ubuntu support list so that people wouldn't feel stupid trying to overcome this difficulty.
The problem you are experiencing is actually not related to the version of Ubuntu. Many of my friends who upgraded from 12.04 to 14.04 assumed that somehow broke you-tube or Firefox, but in fact it was just a matter of timing. I know this for certain because their complaints came in at the same time that I was (and am) still using 12.04, and I was having the same problems.
In my case the ADOBE FLASH driver required to watch both Hulu and You-tube first stopped working under my installation of the "chromium" web browser, and then about a month later Firefox too suffered this change. I was annoyed and several of my Ubuntu friends as well, so after one of them who is too proud to ask for my help took his laptop to Best Buy's Geek Squad and paid them $120 to re-install Firefox... which didn't even help, I decided to see what could be done, before anyone else tried something equally fruitless and expensive.
At that time what I did was to install the "Chrome-beta" web browser (not the original Chromium) and that somehow overcame the problem. If I understand you correctly, the regular Chrome web browser also works, I'll alert my friends.
It is truly unfortunate for Ubuntu that this issue of timing has started the false rumor (which I've been hearing for weeks now) that upgrading to Ubuntu 14.04 somehow makes it impossible to watch Youtube videos.
**UPDATE December 21, 2014: **
I've been wrestling with the idea of deleting this post. It offends me that I received ANY down-votes from people simply because the BETA version of chrome I recommended was out of BETA not long after.
In any case, it seems that the Google Chrome browser worked better for me than the Google ChromIUM browser which was the only version listed in the software center. It is possible that by this date, all browser work equally well but that's not what I continue to hear from frustrated Ubuntu users who turn to this site for help, which is why I'm choosing to update this answer instead of deleting it... for now.
Here is the link I last used to help a friend install the Google Chrome browser on their laptop and permit them to once again view videos.