This is a subtle question of scientific jargon. Indeed, in everyday language, "dynamic" is preferred, and "dynamical" at most is seen as an awkward synonym. My feeling is that "dynamic system" would be the everyday language way of referring to a system that was actually changing in time, moving. Whereas a "dynamical system" in physics, mathematics and at least to some extent beyond is a system that by its nature is capable of exhibiting change in time, i.e., being dynamic; moreover, we make a statement about the reason for that capability, i.e., there is some description of what causes its specific dynamism. One distinguishes between a merely kinematical description and a dynamical description - a kinematical description merely addresses how something moves, whereas a dynamical description addresses why something moves (say, some variational principle).

This is a subtle question of scientific jargon. Indeed, in everyday language, "dynamic" is preferred, and "dynamical" at most is seen as an awkward synonym. My feeling is that "dynamic system" would be the everyday language way of referring to a system that was actually changing in time, moving. Whereas a "dynamical system" in physics, mathematics and at least to some extent beyond is a system that by its nature is capable of exhibiting change in time, i.e., being dynamic; moreover, we make a statement about the reason for that capability, i.e., there is some description of what causes its specific dynamism. One distinguishes between a merely kinematical description and a dynamical description - a kinematical description merely addresses how something moves, whereas a dynamical description addresses why something moves (say, some variational principle).

Indeed, with respect to the latter sentence, there's a clear contrast: A "dynamic description" would be a description that itself changes in time; a "dynamical description" is one that explains the change in time of some other object that is being referred to.