One's for water at room temperature (the white one), the other's for chilled water (the blue one). There are some water coolers that even give you hot water !!!
I guess if you press the blue button then water passes along a tube, through the "chiller" bit of the equipment (which is presumably some sort of heat exchange - like a fridge) and comes out of the tap. The white button must bypass the coolant system cos, as you say there's only one bottle at a time connected up and that's at room temperature.