A method to calculate the classical and quantum free energy of a liquid from a computer simulation by using cell theory [J. Chem. Phys. 2007, 126, 064504] is tested for liquid water and ice Ih against experiment as a function of temperature. This fast and efficient method reproduces reasonably well the experimental values of entropy, enthalpy, and free energy of a liquid across the supercooled, stable, and superheated range of temperatures considered. There are small differences between classical and quantum results of water at 298 K, necessitating a small correction term to reproduce water's enthalpy of vaporisation. Only at higher temperatures is entropy underestimated by up to 9 J K-1 mol(-1) as verified by thermodynamic integration calculations. Satisfactory agreement for ice, however, is only obtained by using the quantum formulation. Even then, at higher temperatures, the entropies exceed experiment by up to 15 J K-1 mol(-1). Further insight into the quantum nature of water is provided by inspecting the temperature dependence of the frequencies. The harmonic approximation is further supported by the harmonic force and torque distributions and the very similar entropies obtained from the force and torque variances. All these results suggest that the single molecule harmonic oscillator approximation for water, although not exact, provides a rapid, insightful, and useful means to evaluate the thermodynamic properties of water from a computer simulation in a way that can account for the quantization of water's energy levels.