Live and learn, I found usecase where lack of animation makes user interaction unobvious and complicated. It's not about going from one square page to another though, but about applying transformation to graphs.
It all started with my desire to study dependencies among Android modules (which is task towards my "Packaged Android" project). I found adepends.py tool (my fork), which is very nice and dumps dependency info in .dot format for rendering with Graphviz. Looking at few graphs it became obvious that for comfortable browsing there should be interactive graph viewer with few capabilities which I summarized in Stackoverflow question.
My searching didn't turn suitable viewer, but fortunately I found xdot.py, which is extremely cool graph viewer app written in Python, which was suitable base for easy implementation of needed features.
And here story comes to the topic of the post - once I implemented prototypes of those features, they indeed proved to be really helpful, but didn't give aesthetic satisfaction - that's because when hiding a node for example, Graphviz's dot completely overhauls rendering of a graph, so user loses track of context he used with. Having shape of a graph morph from old to new state would help a lot.
Actually, animation can help even with simpler graph operation, for example going from one node to another in a complex graph - seeing a ride over an edge which connects them helps intuitive understanding a lot. See ZGRViewer's "link sliding" feature in a video on Youtube:
Simplified form of going from node to node is implemented in xdot.py, except that it moves between nodes via shortest path, which doesn't capture context of curved edges. (Btw, ZGRViewer may be full of cool ideas, but its UI grossly sucks comparing with XDot, literally, it's outbreak from last century). I seriously consider implementing non-linear animation approximating Bezier curves for XDot.
So that's it - going from hating animations to wanting implement non-trivial animations myself in couple of days ;-). (That's also the first time I embed video?)