Dry humor is saying something funny with a straight face. While it can be sarcastic, it often isn't. What's more, all sarcasm isn't dry.
Much sarcasm is not said with a straight face but with a pointed tone that calls attention to itself, particularly when the one employing it wants the joke to be gotten instead of missed or doesn't want to be misunderstood as actually meaning what they are saying, which meaning is by definition the opposite of what they are saying. Dry humorists make no effort to ensure you get the joke, seeming not to care if you miss it. Moreover, someone can say something dryly that is intended to be funny without it being sarcastic.
The "straight man" in comedy is a form of dry humor--he is not innately sarcastic but just the one with whom we identify to whom all of the craziness happens, the Dean Martin, the Jerry Seinfeld. In order to be dry, one need merely remain sane, stoic yet observing, in the face of insanity. Just as often, dry humor is exercised when someone makes uncanny observations of the truth by saying exactly how it is, not by saying the opposite of what they mean.
Amy Schumer, for example, is frequently very dry and very funny by flatly telling the truth of a situation.
Sarcasm is not always mean, as some have said. An example would be if you are making a pie and you need apples and you buy 100 apples, then you say "do you think I bought enough apples?" It's funny.
People who don't understand sarcasm are missing a gene.
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