ANSWER: One means to drink one's fill of something while the other means to drink less than one would like.
입 can mean either "mouth" or "lips," and 목 can mean either "throat" or "gullet," so since 축이다 means "to wet" or "to moisten," 입을 축이다 literally means "to wet one's lips or mouth" while 목을 축이다 literally means "to wet one's throat or gullet." That means that one needs to drink more to wet one's throat than to just wet one's lips and mouth. Therefore, 목을 축이다 implies that one drinks one's fill of something while 입을 축이다 implies that one does not drink as much as one would like.