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Punish and castigate are synonyms. Punish is used more frequently than castigate. Castigate is a fancier word that originates from latin. In spanish you would say "castigar" Penalize ( verb ) penalty ( noun ) is a type of punishment. It is often used in sports as well as other situations. The soccer player tripped the other player and received a yellow card as a penalty. He was penalized. You could use the word punish, but the word penalty is most often used in this situation. You would find the word castigate used more often in literature or other more formal written language. A child would not know what castigate means.

castigate：（正式）严厉责骂，例如：he was castigated for not setting a good example. 他因为未能树立好的榜样而遭到斥责。