We study the dependency of the quantum spin dynamics on the particle number in a system of ultracold spin-1 atoms within the single-spatial-mode approximation. We find, for all strengths of the spin-dependent interaction, convergence toward the mean-field dynamics in the thermodynamic limit. The convergence is, however, particularly slow when the spin-changing collisional energy and the quadratic Zeeman energy are equal; that is, deviations between quantum and mean-field spin dynamics may be extremely large under these conditions. Our estimates show that quantum corrections to the mean-field dynamics may play a relevant role in experiments with spinor Bose-Einstein condensates. This is especially the case in the regime of few atoms, which may be accessible in optical lattices. Here, spin dynamics is modulated by a beat note at large magnetic fields due to the significant influence of correlated many-body spin states.