Lyric discussion by u2nspenserfan 

An organ grinder is a street performer who plays a street organ and, in old times, sometimes had a trained monkey to scamper around with a cup for coins. Now they often use stuffed or brass monkeys in an homage to those days.

That being said, to me, that makes "punish the monkey" means the little guy gets punished, while "let the organ grinder go" means the person with the power gets away with it. I think this could be applied to many things, from criminal enterprise to the government. It often seems the lackeys get punished heavily, and the guys pulling the strings get off with little or no punishment.

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