OK, here is a quick rundown on why GoreTex. If you are bored with technical details, skip to the short version.
Here is a long version: GoreTex (or a similar membrane) is designed to let the water vapor (steam) through and keep water droplets from penetrating the membrane. DWR coating is designed to make water bead on the outer fabric layer to prevent the formation of a thin boundary layer of water on that fabric. Why is this important? becuase GoreTex membrane is a passive object, thus to transport steam it needs some driving force. Normally, the driving force comes from the absolute humidity inside the warm garment being higher than the absolute humidity of the cold outside air (the relative humidity of course goes the opposite way, but that's behind the point). Formation of the water layer on the outer face of the fabric (when the DWR fails) kills that driving force and effectively shuts down the GoreTex membrane for vapor transport. The membrane still remains waterproof, so rain or snowmelt would not penetrate it, but now it has the breathability of rubber (i.e. none) so you will get wet from your own sweat.
Short version: Without a functioning DWR GoreTex remains waterproof, but stops being breathable, so you won't get drenched from rain/snow, but would still get uncomfortable becuase of your own sweat.