A reaction with bacteria produces odor. Aluminum and zirconium salts in antiperspirants react with sweat to create a plug that stops up the sweat gland. They also contain some anti-bacterial agents, as do deodorants, which are mostly perfume. Facial sweating from spicy foods comes from capsaicin, a component in the food that stimulates nerves in the mouth that usually respond to rises in temperature. When comfortable and at rest, the body typically gets rid of about 60 percent of excess heat via infrared radiation.
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.