This is just a personal opinion, of course.  I relate "humid" only with the weather, air, etc.  I relate "damp" with things such as towels, walls, houses, and the weather and air too.

This area is very humid in July.  (Referring to the weather.)

It (the air/weather) is very humid today.  (The air feel so heavy and damp today that it may rain.)

The humidity today is killing me.  (The air feels so heavy with water that it bothers me a lot.)

This area on the wall is damp.  (It feels wet to the touch.  You can see water on it.)

The boys left their damp towels on the floor.  (The towels are not dry.)

The boys left their wet towels on the floor.  (The towels are more than moist or damp, they are wet.  If you squeeze the towels, water will come out of them.)

The house was always damp, and mold was always a problem.  (The air in the house smells and feels wet.)

The dew point is so high that the floors and walls are always damp.  (You can see beads of water on the floors and walls.)