It eats bandwidth, yes, like any animation, but at least the GIF89a format stores only the differences between frames. The other two methods must send every frame afresh. Client pull is appropriate for slideshows and infrequently updated dynamic information. This is a dynamically updated status report for CWRU's campus network. Client pull lets it be updated every three minutes, without leaving a connection open to the server.
Server push is appropriate for dynamically updated information that has to be changed more often.
I've never found a need for it. Finally, GIF89a animation is appropriate for cute little moving pictures. Bruce Mlinar, a student engineer here, did this animation to teach how to plug in a fiber-optic cable.