  We present a comprehensive comparison of spin and energy dynamics in quantum
and classical spin models on different geometries, ranging from one-dimensional
chains, over quasi-one-dimensional ladders, to two-dimensional square lattices.
Focusing on dynamics at formally infinite temperature, we particularly consider
the autocorrelation functions of local densities, where the time evolution is
governed either by the linear Schr\"odinger equation in the quantum case, or
the nonlinear Hamiltonian equations of motion in the case of classical
mechanics. While, in full generality, a quantitative agreement between quantum
and classical dynamics can therefore not be expected, our large-scale numerical
results for spin-$1/2$ systems with up to $N = 36$ lattice sites in fact defy
this expectation. Specifically, we observe a remarkably good agreement for all
geometries, which is best for the nonintegrable quantum models in quasi-one or
two dimensions, but still satisfactory in the case of integrable chains, at
least if transport properties are not dominated by the extensive number of
conservation laws. Our findings indicate that classical or semi-classical
simulations provide a meaningful strategy to analyze the dynamics of quantum
many-body models, even in cases where the spin quantum number $S = 1/2$ is
small and far away from the classical limit $S \to \infty$.
