Before the beta I would've thought questions like this one would be such basic concepts that most contributors would be familiar with them. But after a few questions on gender and animacy it seems that some people do confuse the two even here.
Noun classes and classifiers are similar in one respect: certain lexical roots (like woman, man, tree, etc) will belong to/appear with certain noun classes and not others, and similarly, such roots in other languages will appear with certain classifiers and not others. The operative difference seems to be in the semantic import of the two types of classes. As the discussion here suggested, grammatical gender (or grammatical noun class) is not always systematically tied to language-external properties (like "feminine gender" or "animate class" necessarily picking out things we would ordinarily think of as "feminine" or "animate" respectively), but it is generally not (compositionally) important whether it does or not. Classifiers, on the other hand, are related to the semantics of quantity, be it to specify a type of portion or measure, or the cardinal count of a given nominal, etc. See Borer's work theorizing on the relation between the Chinese classifier system and the semantics of quantity in languages like English. Representations of quantity are quite important to natural language semantics: some of the most important results in this field concern the mathematical properties of natural language quantifiers. I've focused on a semantic difference; the wiki page on classifiers gives more of a morpho-syntactic list of differences.