There is a difference between saying goodbye and letting go.
Goodbye is not permanent. You can meet years later as old friends and share what happened in your life. You can smile and laugh about all the nonsense that you both went through.
However, letting go is being okay with never seeing this person ever again…being okay with never knowing how their life turned out…being okay with fifty or more years of silence… being okay with running into that person at a grocery store and having them not acknowledge your presence.
In move on, you’re talking about a situation in which something has happened and it continues to bother you. Instead of dwelling on it and causing stress or depression over it, the advice is meant to put it behind you and get on with your life.