Having movement on your landing page grabs attention and makes people want to watch, even for just a few seconds. Those few seconds are golden. Your logo animation can be as simple as turning on and off, or as complicated as my little dice. Just imagine it.
This butterfly is an example of a more complicated GIF, using 3D modeling and animation.
A couple of very simple 2D animations yet still quite effective at attention grabbing.
Simple yet effective.
Here is a sample of Aaron Friezen Photography before and after.
Heather Krisher Lantow has this animation on the bottom of all of her professional emails:
Another Email signature line.
Dr. Janet Suh, Healthy Paws Veterinary Care. My
Gina Cervino, Attorney at Law