Screen cap software
- Mac: SnapzProX, ScreenFlow, Camtasia, OBS, iShowUInstant, ScreenRecorderPro
- Win: Camtasia, LICEcap, OBS
- You essentially need screen recording software which will record movies, not simply make images.
Photoshop Extended (although I'm sure many video editing apps would work)
I would also reiterate your comments about using them sparingly. I tend to only do these when there's some aspect which would require 15-20 images to explain and a 10 second animation does it better. I don't add animations for the sake of adding them. Static images are preferred, and only when a static image or text can not convey the proper answer do I move to the animation, and only then if it can be done quickly and succinctly.
2019: In recent years I've had to drop SnapzProX for video captures - it stopped working for video (Still works for static grabs though). So I've moved to IShowUInstant most of the time.