The real factor here is the area of damage. Watermelon is very dense, thus, it will hit in a smaller area causing higher pressure.
While cotton will hit you on a larger area, this pressure will spread and lose its power.
Pressure = Force/Area. Less area will make the pressure go up.
If the cotton was pressed and cubed in a 5inch by 5inch cube (arbitrarily and just for clarification) then the watermelon and cotton will do feel the same when they hit.
Think of it like water in the shower. When you increase the pressure by turning the shower head, what its doing is making water flow throw smaller holes which increases its pressure.
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