On the basis of ab initio Gibbs ensemble Monte Carlo simulations, we map the liquid-vapor phase diagram of water described by the RPBE density functional supplemented by D3 dispersion corrections and estimate the critical point by density extrapolation. Knowing the approximate location of the critical point, two sets of ab initio molecular dynamics simulations at gas-like and liquid-like densities deep in the supercritical phase of water are carried out where particular attention is payed to ergodic sampling in view of large correlation lengths and long correlation times. Structural, H-bonding, and dipolar properties of RPBE-D3 water are analyzed along the liquid-vapor coexistence curve upon moving toward the critical point and compared to those in the supercritical state. The properties of high-density supercritical water are astonishingly similar to those of the liquid on the coexistence curve under subcritical conditions at comparable density. Upon decomposing the molecular dipole moments into purely configurational and electronic polarization/charge-transfer contributions, it is demonstrated that the latter play a decreasing role in liquid water upon approaching the critical point on the coexistence curve. Moreover, these many-body effects are systematically suppressed in supercritical water due to the significantly reduced H-bonding network.