The expression, hold on, is used to describe your grip (or grasp). For example, you can ask someone to hold on to the railing when they are going down the stairs. You can ask someone to hold on to your package while you put your coat on. Both expressions can be used when you want someone to be patient for a few minutes. In a restaurant, a waitress might ask you to hang on (or hold on) when she is rushing to get your bill. When you answer the phone, you can ask someone to hang on (or hold on) while you go to find a pen and paper. Additionally, the expression, hang on, can be used to describe enduring for a longer time. If your friend is going through difficulties, you might tell them to hang on because, in time, their circumstances will get better. Hope this helps.

They are synonyms. Both are an informal request to wait for a short amount of time. "hang on" used to be considered as less formal than "hold on" but that difference has faded over time.