Usually Internet Explorer is able to display most videoclips because it has a native support for Flash Player. It takes advantage of the system's libraries to show the Flash player on a website. Mozilla Firefox needs Adobe Flash Player ActiveX installed in order to display flash content or anything related to flash multimedia.
If you want to install this ActiveX, please download the package, close all your browsers and then perform the installation. There is no need for a system restart because the changes are instant.