I am redesigning a site, and the client is set on keeping a small looping Flash animation that is on the current version.
They have asked if it would be better to replace it with an animated GIF version or to keep the current version, and I don't know the answer (apart from get rid of it!). Is either method preferred?
Animated GIF, without a doubt. Much easier to code and maintain.
GIF is very old, and every spam advertisement out there today are using Flash.
Use GIF on mobile and to keep bandwidth low. It is also easier to use an img tag with gif's.
Use Flash if you want better animation control, but it could be blocked by Anti-Ad blockers.
Flash, but use redundancy. Using a library such as SWFObject, you can dynamically load a flash file on top of another element (also called "alternative content"). There is a good tutorial on this which you can find here (it is also listed on the SWFObject website).
So, why flash and not gif? Essentially, this is because of the alternative content solution described above, as you get more versatility in what content you deliver. Mobile users, for example, are going to appreciate a static image (or nothing at all) instead of an animated gif.
Having said that, you will also be able to deliver better animations with flash. And, under certain circumstances, you'll be able to match (or even better) the .swf filesize with the .gif for the same animation.