Which is more wet: ‘moist’ or ‘damp’?

Solution 1:

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

Solution 2:

As a counterpoint to the definitions that appear in Fard's answer (by way of Longman's Dictionary, I offer these definitions from Merriam-Webster's Eleventh Collegiate Dictionary (2003):

damp adj. {ME, black damp, fr. MD or MLG, vapor; akin to OHG damph vapor} (1590) 1 a archaic : being confused, bewildered, or shocked : STUPEFIED b : DEPRESSED, DULL 2 : slightly or moderately wet : MOIST {a damp towel} also : HUMID {damp weather}

...

moist adj. {ME moiste, fr. AF, perh. fr. VL muscidus, alter. of L mucidus slimy, fr. mucus nasal mucus} (14c) 1 : slightly or moderately wet : DAMP 2 : TEARFUL 3 : characterized by high humidity

The Eleventh Collegiate also includes this usage note under the entry for wet:

DAMP implies a slight or moderate absorption and often connotes an unpleasant degree of moisture {clothes will mildew if stored in a damp place}. ... MOIST applies to what is slightly damp or not felt as dry {treat the injury with moist heat}.

So the main entries for the two words give the relevant definition of damp as "slightly or moderately wet : MOIST" and the relevant definition of moist as "slightly or moderately wet : DAMP." Those are about as close to interchangeable definitions as you can find in Merriam-Webster. The usage note comparing the two words does note that some senses of damp involve unpleasant or uncomfortable settings, but it doesn't draw a sharp, categorical contrast between unpleasant wet (damp) and pleasant or suitable wet (moist).

In U.S. usage a damp towel is neither more nor less pleasant than a moist towel—and many other instances exist where damp and moist might be used interchangeably without significantly different connotations. UK usage may be different, but for the United States I don't think that a definite difference in wetness is implicitly understood as between damp and moist.