"Facetious" comes from a Latin word that means "jest." A facetious comment is a joking comment—often inappropriate. Think of a jester or joker making a funny face at you, and remember the first part of "facetious."
"Sarcastic" comes from a Greek word that means "to speak bitterly or sneer." Ouch! A sarcastic response if less funny than a facetious response and more bitter and harsh. Think of a sarcastic person sneering at you, and remember that both words start with the letter "s."