@sjb drink implies alcohol, usually hard liquor or mixed drink; beverage is not common when talking, but when used often you're being discreet on purpose for whatever reason, so it can be sort of a euphemism for a stiff drink, or specifically obvious as in, "I need an alcoholic beverage", or "I need a caffeinated beverage". You would never really hear, "I'm having a caffeinated drink" unless it was alcoholic.
@sjb Yeah, soft drink directly means no alcohol, but there's not really such a thing as a soft beverage. Coffee and tea are caffeinated beverages. I guess it depends a little on context, too. Sports drinks are pretty specific and non alcohol. Anyway, "come over and have some drinks with us" definitely does not mean "come over and have some tea".