Honeycam is a powerful tool for building animated GIFs, WebP or WebM images. It's a commercial product, but may be used for free with some restrictions (max 10 seconds recording time and a small watermark added).
The program includes an easy-to-use screen recorder. Place a rectangle around the area you'd like to record (a video player, for example) and you can capture it with a click.
Honeycam is also able to build animations from a series of still images, or you can open an existing animated GIF/ WebP directly.
The highlight of the program is Honeycam's editor. There are tools to adjust the brightness and contrast of individual frames, apply special effects, crop or resize them, add images or text watermarks, set transitions or adjust animation speeds.
You get various fine-tuning options, including the ability to delete selected frames, copy or move them around, or set individual frame delays. And if the file is too large - a common problem with GIFs - a "Reduce" option deletes either all the even or the odd frames, cutting it down to size while preserving the original speed.
Version 1.02 is a maintenance release.
It's expensive, but Honeycam does have a good set of recording and editing options. The watermark on the free version is minimal, too, so if you can live with its 10 second time limit then it might be worth a try.