Skylight filter is similar to a UV filter, instead, it is on a slight pink tone coating. This removes the excessive blue hue in the atmosphere, gives a better balance in picture between the red and blue tones.
The UV, Skylight and Haze filters works better with film which are more reactive/sensitive to UV lights and tonal changes (digital has its WB to be changed).
As said, polarisers polarise the scattered lights around the atmosphere, creating a high contrast image. Sometimes the effect is so high that the sky will appear dark blue (as seen in some wonderful landscapes).
It not just darken the sky and increase contrast, it also removes reflection from non-metallic surfaces. Specially with water.
Samples:
Without CPL:
With CPL:
Full polarised sky and contrast boost: