Historically, mist was categorized as a meteorological or atmospheric condition. Hence, Webster's Dictionary of Synonyms (1942) lists mist in a category with haze, fog, smog, and brume:
Haze, mist, fog, smog, and brume agree in denoting an atmospheric condition which deprives the air near the earth of its transparency. ... Mist applies to a condition where water is held in suspension in fine particles in the air, floating or slowly falling in minute drops.
Figuratively, as Merriam-Webster's Eleventh Collegiate Dictionary (2003) observes, mist can also refer to various things suggestive of atmospheric mist:
4 a : a cloud of small particles or objects suggestive of a mist b : a suspension of a finely divided liquid in a gas c : a fine spray
In contrast, though vapor can be used to describe mistlike conditions, it has a stronger connection to gaseous rather than particulate solid or liquid states. It comes from the Latin word vapor meaning steam. Again from the Eleventh Collegiate, here are the first three definitions:
vapor n (14c) 1 : diffused matter (as smoke or fog) suspended floating in the air and impairing its transparency 2 a : a substance in the gaseous state as distinguished from the liquid or solid state b : a substance (as gasoline, alcohol, mercury, or benzoin) vaporized for industrial, therapeutic, or military uses; also : a mixture (as the explosive mixture in an internal combustion engine) of such vapor with air
Given the association of vapor with heating or sublimating into lighter than air gases (and leaving aside the stubborn fact that vapor has long been used as a way to describe fogs and—inevitably—mists), one might argue that a vapor in many instances rises, while a mist tends to descend slowly.
Incidentally, Noah Webster, in his Compendious Dictionary of the English Language (1806) gives only the heated or sublimated sense of vapor:
Vapor n. a fluid rendered volatile and elastic by heating, fume
Samuel Johnson, Dictionary of the English Language (1756) has a more varied (and interesting) take on the word:
VAPOUR. s. [vapor, Lat.] 1. Any thing exhalable ; any thing that mingles with the air. Milton. 2. Wind, flatulence. Bacon. 3. Fume ; steam. Newton. 4. Mental fume ; vain imagination. Hammond. 5. Diseases caused by flatulence, or by diseased nerves ; melancholy ; spleen. Addison.