Temperature is the main regulator on evaporation, that and relative humidity. Distilled water vs boiled water you won't notice any difference without very precise equipment. Standing water vs boiling hot water you would see a major difference... After all don't you see steam coming off of something hot? That is water vapor leaving due to heat...
Well your question has been debated by many scientists and they have reached a conclusion but different people have different opinions and views. Personally I believe that boiled water evaporates faster then normal water because boiled water after it has been cleansed has more simpler molecules and parts and will evaporate faster then tap water. Furthermore it also depends on the temperature of the water. Hope this answers the question.
I think boiled water evaporates faster because its already hot and can skip the process of heating unlike tap water
Distilled water has already removed the minerals that slow down the process of boiling, so I would say that undistilled would be slower. Great question!!