They both mean the same thing, and neither is really appropriate in this context. When you bow out of, or step aside from, a role or position it means that you are leaving or resigning from it. In this case, the musician is cancelling a future engagement, rather than leaving.

This might be a difference between British and American English, but your first example with "bow out" sounds okay to me. Although the term does usually mean to resign from a position, it can also mean "to retreat or withdraw" (as defined by the Oxford English Dictionary: "to retire (gracefully); to retreat or withdraw; to resign"). You normally can't "withdraw" from something before it begins, but I think in a case where there was a prior long-term commitment (the singer had already agreed, planned, and intended to do the charity event), it's possible to describe him/her as "bowing out" of the commitment before the event itself takes place. As Fergus notes, the phrases "back out" and "duck out" have negative connotations, whereas "bow out" suggests that the person leaves gracefully. The term "bow out" therefore seems preferable to me, if you want to leave a more positive impression of the singer.

I agree with Jimmy. To "bow out" means to resign from a job/role/position that you have performed for some time. For example: "The current chairman of the club will bow out at the end of this month" To "step aside" means to resign from a position, usually to let a new/junior/less experienced person take it. For example: "At the age of 36, the captain thinks it is time for her to step aside and let new talent come through". Neither example is correct in this context. I might use: "The famous singer backed out of the charity concert at the last minute" "The famous singer ducked out of the charity concert at the last minute" both of which have negative conotations for the singer. P.S. If this is an exam question, I imagine that the "correct" answer is 1, because singers usually bow at the end of a concert. However, it is not correct in English.