Heat travels from the skin to cooler surrounding objects. It's when the body heats up that sweat and evaporation become more vital. The body stays hot because the air is already thick with water vapor. There's little place for evaporated sweat to go, so the rate of evaporation slows. Since it's evaporation and not sweat, per se, that cools the skin, the skin stays hot and wet.
Danger is at hand if someone suddenly stops sweating in hot weather.
That means the body is in shock, preserving water and going into possibly fatal heart stroke. Some people are born without working sweat glands. The condition, hypohidrotic ectodermal dysplasia, is rare, inherited and incurable. It also produces dry skin, wrinkled skin, sparse dry hair, small brittle nails and cone-shaped teeth. Reduced activity is the most common treatment.