Both have the same meaning except that "admit" has a connotation that you had done something wrong or guilt you have. "admit to the truth" - you have to acknowldge the truth because you were once wrong, failed to realize it significance etc.

In this sentence, either is fine. However, "admit" normally means to know something is true and then tell someone else. "Acknowledge" means "admit" or "recognize". You can "admit" only the truth. You can "acknowledge" the truth, or lots of other things, like people.

"Admit" would imply that she actually said "Ok, I have a mustache." "Acknowledge" can just mean just to accept a fact, although it also can imply that she verbally admitted it, as well. The meaning is similar and the difference is subtle, so don't worry too much about getting them mixed up. "Admit" is much more common in conversation, with "acknowledge" being more formal or literary. You might hear it more commonly between two radio operators: "Are you receiving this message? Acknowledge." "(The message is) Acknowledged."