Classical simulations of simple water models reproduce many properties of the liquid and ice but overestimate the heat capacity by about 65% at ordinary temperatures and much more for low temperature ice. This is due to the fact that the atomic vibrations are quantum mechanical. The application of harmonic quantum corrections to the molecular motion results in good heat capacities for the liquid and for ice at low temperatures but a successively growing positive deviation from experimental results for ice above 200 K that reaches 15% just below melting. We suggest that this deviation is due to the lack of quantum corrections to the anharmonic motions. For the liquid, the anharmonicities are even larger but also softer and thus in less need of quantum correction. Therefore, harmonic quantum corrections to the classically calculated liquid heat capacities result in agreement with the experimental values. The classical model underestimates the heat of melting by 15%, while the application of quantum corrections produces fair agreement. On the other hand, the heat of vaporization is overestimated by 10% in the harmonically corrected classical model.