My favourite style is talking about seemingly unrelated things first and then answering the primary question at last. In that spirit, here goes the first unrelated question.
Why do we sweat?
Normal human body temperature is 98.6 °F. When the external temperature drops or increases, it changes our body temperature too (obviously), but our body works to bring it back to the normal temperature. When it's cold, our body shivers to generate heat; when it's hot, our body sweats to cool itself down.
How exactly does sweating help?
You need to know the physics of evaporation to understand that. Take a very hot metal plate and pour water on it. You'd see water boiling, and soon all of the poured water would disappear. If you measure the temperature of the plate now, you can see that it's become colder than before pouring water. What's really happened is, water took the heat energy from the plate and used it to become water vapour (which then mixed with air in the atmosphere).
Sweating is pretty much the same. Our body becomes hot because of the weather. As a response to it, we secrete sweat all over our body. Sweat then uses the heat from our body to evaporate. Since some body heat is expended on sweat evaporation, our body cools down. (In a way, we return the unwanted heat back to the environment.)
Humid weather
What happens in hot-humid weather? Because it's hot, our body secretes sweat to cool itself down. Now we get all sweaty, but nothing happens after that. Why? When a liquid becomes gaseous vapour, it needs someplace to go to: usually it mixes with the air in the atmosphere. But there's only so much air around us and only so much water vapour can mix with it. In dry weather, the air can take in a lot of water vapour because it doesn't have much in it already... it's dry. But in humid weather air is pretty much "full" so the vapour intake will be much less. The sweat stays on our body until we wipe it off. And our bodies remain hot.
In dry weather we probably sweat the same amount, but we don't notice it because the sweat evaporates pretty much immediately. Our bodies don't get too hot either.