there are some subtle differences in usage The root meaning of wait is simple, has no context really - 'don't act, until an event occurs' the root of 'hang on' is more dramatic. literally ' i know that you are about to fall from where you are hanging on to (a cliff perhaps) and I am coming to assist you!' So you could say 'wait' is simply a request. Hang on means 'i understand you need me there now, I'm trying to be there as soon as I can.' Hang on is also used very casually as an interjection, to imply surprise or drama: e.g. 'Hang on, are you telling me that she is dating him now?'

"Wait" is straightforward, simple, and often used. It is neither formal nor informal. It's a simple command to wait for an unspecified amount of time. To make it polite, you say "please wait." "Hang on" is informal. It carries the idea that the speaker _really_ wants you to wait, that they've been interrupted by something urgent, and that it won't be a long wait--they'll be back to you soon. "Hang on, I need to take a call from my wife but it won't take long, I'll be right back."