The sweat is not cold as it comes out of our body. It is body temperature, and is only cooled through evaporative cooling. I just looked up the numbers to be sure, and for each kilogram of water vaporized 2,257kJ of energy is used. Thus, your skin feels cold.
Evaporative cooling works great and is also used to cool houses.It was very popular where I lived in the Sahara (people sprinkled water on the ground in the patio) and in some parts of the US it is called a swamp cooler.