"let bygones be bygones" is a way of saying let's leave what has happened between us in the past, in the past. For example, if you had a disagreement with someone thta resulted in you not talking for a period of time but you then wanted to start talking again and put the disagreement behind you, you might say to the person "let's let bygones be bygones and make a fresh start" "forgive and forget" - you are telling the person that although they have done something wrong to you, for example, a player on another team has injured you deliberately in the past, you are telling them you forgive their wrongdoing and are willing to forget it happened. You are willing to leave it in the past and there are no hard feelings. Both phrases have the same result, you are putting any previous bad feelings behind you and are willing to start agin. Perhaps a subtle difference between the two is that the first phrase suggests the speaker is not saying one person or the other was at fault but that both have made mistakes or done something wrong whereas, in the second phrase, the speaker is saying that it is the other party who has done something wrong but they are willing to forgive them for it and put it in the past. Both situations are such that the person saying the phrase is "holding out an olive branch" - offering peace between the two. Hope that helps!

Basically. yes. They both mean - allow the unpleasant things that have happened in the past to be forgotten.