The video appears to be an mp4 (the network monitor says so) and it should work fine (it does here). Please ensure you have the latest FF version, available from https://www.mozilla.org/en-US/firefox/all/ (in case the update fails).
I have Firefox 26.0 installed, it still fails on both Windows 7 and Ubuntu 13.10 (64 bit both).
Can't help you with Windows, but I'm interested to see what happens under Ubuntu. It's on my todo list for tomorrow (by date, not by priority).
It appears to work just fine in FF26 under Ubuntu 3.10 without anything extra installed. Have you tried turning it off and on? Try a new profile (firefox -profileManager) or Restart with Add-ons Disabled (help menu), or perhaps it was just a fluke. Could not reproduce (relevant XKCD: http://xkcd.com/583/ ).