Put simply, Animatronics are more complex and programmable, where Animated Figures are simpler and mechanically driven.
Compare a figure like The Auctioneer in Pirates, whose motions are dictated by digital inputs that could be altered to design a different performance for the figure (they used to be controlled by Audio input, hence the name Audio Animatronics), versus something like a Jungle Cruise Elephant, whose individual movements are more cyclical and tend to loop based off the motors and mechanics to create movements that are intentional but not necessarily timed to run together in exact sequence.
Basically, if the performance is the result of carefully crafted programming then it's an Animatronic, but if the performance is the result of carefully crafted mechanics then it's an Animated Figure. It's not always easy to tell the difference just by looking, since Attractions will sometimes use both within the same scene (Little Mermaid uses Sebastian and Ariel Animatronics in the Under the Sea scene, but all the other characters in that room are Animated Figures that mostly have simple rocking motions to them).
Now, Animated Figures can have some pretty complex motions, so it's not automatic that they're worse than an Animatronic, they just tend to move in ways that are more general rather than specifically timed. One of the biggest tells is that if a character is accurately lipsynching it pretty much automatically has to be an Animatronic, since Animated Figures can't do that given how they're built.