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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).