The user experience is miserable. They take longer to start, impossible to stop, burn bandwidth and CPU. Scroll through something like kotaku sometimes, it's an utter poopfest.
300ms is a good target for a page load, not for a pointless animation. And that's a 3MB gif. The page also contains a 2.6MB one, two 6+MB ones and one over 7MB. The entire page is over 30MB. Scrolling through it takes the sort of bandwidth typically used for a high quality streaming movie. At 1 to 2 MB/sec you can easily miss something is an animation in the first place because it won't have time to load. Of course, you're not really missing anything other than a low-fps repeated distraction you can't choose to start or stop. This is a good user experience the way a fresh pile of manure is warm shelter.
All true, but those aren't necessarily problems with the concept of a gif, and could be fixed with another format, that isn't necessarily a full on video.
I think the concept of gif's is fairly great, but the actual format itself has tons of issues.
It seems to be covered adequately by formats supported by current browsers. Try pasting a link to a gif in Facebook. Facebook will convert it to an mp4 and put a big 'GIF' button on top of it. This doesn't seem to bother or be noticed by (statistically) anybody.