damp:
slightly wet, often in an unpleasant way:
moist:
slightly wet, especially in a way that is pleasant or suitable
The same soil would be moist and damp at the same time: moist for planting seeds, and damp for sitting on.
This is the Word Choice note given by Longman Dictionary:
WORD CHOICE:
damp, moist, humid
Use damp especially to say that something is slightly wet in an unpleasant way:
• The room was cold and damp.
Use moist to say that something is slightly wet in a pleasant way or in the way it should be:
• She took a mouthful of the delicious moist cake.
• rich, moist soil
Use humid to talk about the weather or the air when it is slightly wet and makes you feel uncomfortable:
• the hot humid atmosphere of a greenhouse
Source: Longman Dictionary