The two are similar but need to be used carefully in the right context. Back down - is related to being challenged by someone or something. When someone backs down, they essentially give up or surrender. This could be because the challenge is too great or maybe they are fearful. Example: 'The professional boxer challenged the young boy to a fight in the street. However, the boxer was too strong so the boy decided to back down' Bow out: Relates retiring or to a situation where the person has decided STOP taking part. It could be an activity like a sporting game or a business venture. For example: 'after months of negotiation, the businessman decided the company was not profitable and told the Directors he would bow out' Bowing out happens because a person decides to leave the situation or an agreement.