Just a quick comparison between three more modern alternatives to the GIF format for creating looping, animated images. This is by no means comprehensive, but it does illustrate the advantages of using a real video codec over either of these image formats.
30.7 MB — Firefox (mobile), Safari (mobile)
22 MB — Chrome (mobile), Android Browser
Lossless h.264 (4:4:4):
13.9 MB — Chrome
Lossless h.264 (4:2:0):
9.8 MB — Chrome
As you can see, it's no contest. Even at full RGB sampling, h.264 destroys the competition, and with normal 4:2:0 subsampling, you can shave an additional 25% off that! All completely lossless! Of course, that also comes with the worst support among all three formats. Only the desktop version of Chrome could handle h.264's lossless mode. I guess you can't have everything.