We used to be able to use an appliation called Debabilizer to analize each frame of a video clip and generate a Super Palette of 256 colors that best represents all of the frames and could be loaded in Photoshop in the Save for Web (Legacy) dialog box. It would be amazing if Animated PNG would become common as it supports millions of colors.
Of course, we're used to seeing video at millions of colors which usually looks great. For video to look okay, we want at least thousands of colors (16-bits total, 5-bits per color channel and 1-bit for an alpha channel), but that's still in the ballpark of VHS quality from the days of analog video. But when that same 8-bit Color Table needs to be used across 24 or 30 still images (or frames) per second, there's not too much that can be done to improve picture quality. The GIF file format supports a maximum of 256 colors (8-bits for all three colors channels) and for a still image image this isn't so bad.
