We investigate the accuracy provided by different treatments of the exchange and correlation effects, in particular the London dispersion forces, on the properties of liquid water using ab initio molecular dynamics simulations with density functional theory. The lack of London dispersion forces in generalized gradient approximations (GGAs) is remedied by means of dispersion-corrected atom-centered potentials (DCACPs) or damped atom-pairwise dispersion corrections of the C6R-6 form. We compare results from simulations using GGA density functionals (BLYP, PBE, and revPBE) with data from their van der Waals (vdW) corrected counterparts. As pointed out previously, all vdW-corrected BLYP simulations give rise to highly mobile water whose softened structure is closer to experimental data than the one predicted by the bare BLYP functional. Including vdW interactions in the PBE functional, on the other hand, has little influence on both structural and dynamical properties of water. Augmenting the revPBE functional with either damped atom-pairwise dispersion corrections or DCACP evokes opposite behaviors. The former further softens the already under-structured revPBE water, whereas the latter makes it more glassy. These results demonstrate the delicacy needed in describing weak interactions in molecular liquids.