In Safari on OS X, Facebook does use the native <video> player.
I'm guessing that Facebook encodes video h264 which isn't natively supported in Firefox; rather it relies on support in the operating system. I'm not sure if Chrome on Linux supports h264, however since Chrome also includes its own Flash player I guess that Facebook may be using their own flash player anyway.
Pretty sure there are some settings you need to enable in about:config to get H264 working in Firefox (assuming you have the right gstreamer stuff installed).
