Density functional theory (DFT) within the generalized gradient approximation (GGA) is known to poorly reproduce the experimental properties of liquid water. The poor description of the dispersion forces in the exchange correlation functionals is one of the possible causes. Recent studies have demonstrated an improvement in the simulated properties when they are taken into account. We present here a study of the effects on liquid water of the recently proposed semi-empirical correction of Grimme et al. J. Chem. Phys. 132, 154104 (2010)]. The difference between standard and corrected DFT-GGA simulations is rationalized with a detailed analysis upon modifying an accurate parameterised potential. This allows an estimate of the typical range of dispersion forces in water. We also show that the structure and diffusivity of ambient-like liquid water are sensitive to the fifth neighbor position, thus highlighting the key role played by this neighbor. Our study is extended to water at supercritical conditions, where experimental and theoretical results are much more scarce. We show that the semi-empirical correction by Grimme et al. improves significantly, although somewhat counter-intuitively, both the structural and the dynamical description of supercritical water.