You have gif with all its quality loss (256 colors and etc.), and then to make matters worse it's getting shoved through webm/mp4 encoding.
And then there's everyone from Imgur to Twitter missing the point of why gif is even a thing. Because it is simple. Because gif is treated as an image and not a video format. Because you can right click it, and save it for later use. Can't do that with "gifv" (seriously, I had to use curl just to get the video file from Imgur), and definitely can't do that with Twitter.
Neither webm nor mp4 can kill gif yet - mp4 is a patent mess, and webm comes closer (4chan for example has a quite big SFW community of people trading videos), but there are issues with Apple refusing to support it, and lack of hardware support.
A small note on a common misconception - the mp4 format is not something that is patented - that's h264.
Broadly speaking video files are first a container (e.g. mp4, avi, mkv, ts, etc). A container is sort of like a zip file. It groups together metadata about the file as well as the audio, video and possibly subtitle streams. In fact if you view an mp4 file at this level it reads like binary XML with 4 byte tag names.
The video streams are encoded using a video codec - e.g. h264, h265 (HEVC), vp8, vp9, ProRes and many others. Audio codecs are similarly varied - e.g. AAC, mp3, vorbis, theora and others. These are the parts of the file that can be patent encumbered.
In theory there is nothing stopping you from making an mp4 with vp8 video and vorbis audio. You can even ask ffmpeg to do it - it will happily oblige (and ffplay will play it). However the reality is that most non-smart video players will freak out at this kind of file.
Most simple players will expect mp4 to contain h264/AAC. This reality is starting to break with h265 encoded video in an mp4 container. Hopefully some of these assumptions will disappear over time. I believe mkv is currently the only reasonably popular container format that can usually play whatever audio/video is inside (mostly because only "advanced" players can even read it).
To add some confusion WebM is actually a hybrid. WebM is a more restricted mkv container with a VP8 OR VP9 video codec and vorbis or opus audio codec.
Try to save a webm if it serves you an mp4, though. Maybe they changed it lately, but I just could not figure it out - .webm and .mp4 both just silently redirected to .gifv last I tried.
Saving a video off imgur works about half the time in firefox, based on ???. There's a bug for it but it hasn't gotten much attention. It's a problem of imgur playing games with headers, embedding a video inside a web page and using the exact same URL for each. I haven't had any issues in chrome.
Twitter's just putting dumb empty divs over the content. And they make it impossible to get the source gif even in cases where the video version is larger and lower quality. Thanks twitter.
Some sites go to quite some length to ensure that users cannot download videos easily (I believe youtube has some javascript that assembles a lot of different video files while playing to ensure you can't get a direct link to the video) but you can go to similar length to protect normal images.
Imgur automatically converts larger GIF files to "GIFV" (WebM or MP4). The size difference is nuts: their example is a 50MB gif being reduced to 3.5MB
Their announcement post about it: http://blog.imgur.com/2014/10/09/introducing-gifv/