Colloidal suspensions are often argued to be an ideal model for studying phase transitions such as crystallization, as they have the advantage of tunable interactions and experimentally tractable time and length scales. Because crystallization is assumed to be unaffected by details of particle transport other than the bulk diffusion coefficient, findings are frequently argued to be transferable to pure melts without solvent. In this article, we present molecular dynamics simulations of crystallization in a suspension of colloids with Yukawa interactions which challenge this assumption. In order to investigate the role of hydrodynamic interactions mediated by the solvent, we model the solvent both implicitly and explicitly, using Langevin dynamics and the fluctuating lattice Boltzmann method, respectively. Our simulations show a significant reduction of the crystal growth velocity due to hydrodynamic interactions even at moderate hydrodynamic coupling. This slowdown is accompanied by a reduction of the width of the layering region in front of the growing crystal. Thus the dynamics of a colloidal suspension differ strongly from that of a melt, making it less useful as a model for solvent-free melts than previously thought.