I found a web map on the New York Times showing early spring conditions throughout the US. I've seen a much more complicated time-slider widget that can be used with ArcGIS Online web maps, however, I really like the simple lines in this map. What technology was used to create this GIF web map?
Don't know what the NYT uses, but Photoshop can create a GIF using its Animation toolbar from a stack of images in separate layers in the same file.
If you need an open source option, GIMP can do that. I did not try creating an animation using GIMP before, but this tutorial of How To Create An Animated GIF From An Image Sequence Using GIMP shows the steps of doing what you want.