They don't though. At similar quality and framerate, a video is much smaller than a GIF (so the radios can be shut down faster) and requires less power to playback (because it's offloaded to the hardware acceleration instead of being implemented entirely on the CPU).
Your average smartphone can play 1080p video without breaking a sweat, not so with 1080p gifs. Hell, a laptop will have permanent high-CPU usage on 1080p gifs, as a video it's background noise (https://www.reddit.com/r/osx/comments/43rrf0/pixel_art_gif_a...)
Why do you hate my phone battery so much?