In a comment to the top answer of this question Darij Grinberg says that

the problem with the dynamical perspective is that it is way harder to grasp for algebraic/combinatorial-minded people than any formula, however complicated it is. I still don't get the difference between a transformation of points and a transformation of coordinates; for me, they're all endomorphisms of a vector space.

Since apparently I'm also an algebraic minded person - I neither can see a difference between those transformations and also view only as endomorphisms - I would very much like to know what their difference consists of (even if the difference manifests itself only on the level of intuition and not of formal mathematics).