We consider the diffusion of a particle in a polymeric hydrogel in the presence of steric, electrostatic, and hydrodynamic interactions. We in particular investigate hydrodynamic effects by comparing Brownian dynamics simulations with and without hydrodynamic interactions. Inclusion of hydrodynamics improves agreement of the diffusion constant between simulations and published experimental data and yields quantitative agreement for gels with electrostatic interactions. Repulsive particle-gel interactions decrease the effect of hydrodynamic interactions on the particle diffusivity compared to purely steric gels, while attractive particle-gel interactions strongly increase hydrodynamic effects. This asymmetry is due to spatial particle-gel correlations.