Fast & Easy. Not entirely intuitive, especially since the screen samples appear to be from a phone. But I just click the Download box under a Twitter video, then select the auto option. That presents me with all the available screen resolutions. Pick one, and the video is saved into FireFox's download folder. Super easy.

Thank you PBION!

Adds a button for every media element (videos, images). Clicking opens a new page offering download options. You have to select quality and then it downloads. It always mentions that some rights of the addon has been changed and I should be reinstalled. The mention is a big red bar on the top of your screen. Following the procedure returns normal use however you could wait, it seems to go away and then it returns. So two clicks and a page switch followed by a reinstall question. + I could download everything I want. Neutral: the procedure. -- The need to reinstall. I'm not running the latest firefox so it might be self inflicted at this point.

It not really an addon. I just links to a website. By the way even that website is useless as it does nothing except showing ads.

Why it disappears and again we have to again go to mozila addons to again refresh it.

Works great, except the download buttons are way too big, to the point that I usually disable the add-on until I need it. Kinda defeats the point.
I like that the filenames of the downloads are based on the text in the tweet (as opposed to just being some random characters).

Files saved from Twitter are recognized by the security system as containing a threat and potentially dangerous. My own files, which I uploaded to Twitter from another computer, when downloaded from Twitter through another system, become "unsafe" after they are downloaded by this application.