Why doesn't the vapor pressure of water depend on the humidity?
The partial pressure of water vapor is equal to the equilibrium vapor pressure at the temperature of the liquid only at the surface of the water.  Above the surface of the water, the partial pressure of the water vapor decreases with distance, and far from the surface, approaches the partial pressure consistent with the relative humidity of the room air.  So there is a concentration gradient of water vapor in the air above the water surface.  Based on Fick's law of diffusion, this provides a driving force for diffusion of water away from the surface.  We call this diffusive flux of water vapor evaporation.  It is driven by the difference between the equilibrium vapor pressure of water at the surface and the partial pressure of water vapor in the bulk of the room air.