Dampness is a condition of being a little bit wet. When your parents pick you up after a pool party, the dampness of your swimsuit might make riding in the air-conditioned car uncomfortable.
If there’s dampness, there’s some amount of moisture. After you cut across her recently watered lawn, your friend may see the dampness of your sneakers and ask you to take them off by the front door. Your mom might have a dehumidifier in the basement to combat the dampness down there and prevent mold from growing.