The results of classical and quantum simulations of liquid water over a wide range of temperatures are compared to probe the impact of quantization on the properties of liquid water. We show that, when treated quantum mechanically, water molecules have an enhanced probability of accessing nontetrahedral coordination in the local three-dimensional structure. We discuss how this enhanced probability, also called "effective tunneling", is related to the dynamics of the hydrogen-bond breaking and molecular diffusion in the liquid. We explore in detail how local molecular environments affect the manifestation of quantum effects and identify a previously unreported and apparently unique behavior of the quantum mechanical uncertainty of the water molecule as a function of temperature. The nonmonotonic behavior of the quantum mechanical uncertainty with temperature is shown to be due to the notable strength of the water-water interaction in the condensed phase and becomes further evidence of the importance of the water structure in the properties of this ubiquitous liquid.