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.