The two expressions not in the wrong and not wrong mean the same thing, and can be used interchangeably in most contexts. However, not in the wrong has the connotation of not having responsibility for something or not having made a mistake. It is usually used in reference to people.
Ex: I'm not in the wrong this time. It was your fault.
Ex: You were in the wrong this time, so you should apologize to her.
Meanwhile, not wrong can be used for the above meaning in addition to describing something that is inappropriate or wrong for a situation.
Ex: This method is not wrong, but it is not efficient.
Ex: You're not wrong. I think this method could work.