I would say the mobile method is easier and more convenient as the complete video downloads. But if you want a higher quality video then I figured out how to do that. Unfortunately in the standard Facebook interface the video and audio play separately, so you'll be downloading each as such.
Find a video on Facebook and open the developer tools by pressing F12 and click network. Start playing the video then filter the network by video.
You'll see urls for both the video and audio, then to download both you open the url in a new tab. Then you need to remove the part of the url at the end and press enter. I'll show a screenshot below, the part which isn't highlighted. When this is done you right click on the video or the audio and save video as or save audio as. Choose a name and directory then click save.
Play the files together in vlc media player or merge them using tool built into vdh.