I not exactly sure what the difference really is, but would believe that there is something to do with origin.
There is this post by another member that may or may not give a hint.
http://www.penturners.org/forum/1390483-post6.html
Damascus referred to is that of Damascus steel, and a wiki read seemed to imply that there really isn't a definitive answer as to what it truly is, and that we tend to "use" the term. Nowadays it's a description of patterned steel (often used in cutlery, in the past used in firearm barrels as well, and recently also used on shotgun barrels).
A wiki of Mokume-gane gives a translation as, "burl metal", indicating a metal patterned to resemble wood grain.
Both are done by using laminates of differing metals. Common understanding in the cutlery industry is that after folding (or however they form the pattern), the blade is then etched with some chemical (the term, "acid" has been mentioned) to obtain the contrast.
My understanding is that it's patterned metal, however I've also seen it used (and therefore used it myself) to refer to polymer clay, and the wiki indicates something with laquerwork.
ETA:
Never did understand the difference the M3's company applied to the references. I thought Damascus refereed to steel, and that Mokume was just a broader reference.