What is the difference between "hang on" and "hold on"?

"Hang on" and "Hold on". I often hear these words, especially on TV. People use them when they want someone to wait for something.

In the provided context, they are very similar.

Hang on:

informal wait for a short time: hang on a minute—do you think I might have left anything out? *(on the telephone) remain connected until one is able to talk to a particular person.*

Hold on can mean the same in general conversation:

[often in imperative] wait; stop:
hold on a minute, I’ll be right back!

While the end result is the same, when used as part of telephonic conversations, hold on often refers more to being placed on hold:

waiting to be connected while making a telephone call:
‘I’ll just see if he’s free,’ Rachel said, and put me on hold

As the ODO page will confirm, there are a number of other idiomatic uses of hold (hold your horses, hold your fire, hold it, etc.) which share similar connotations of stop or wait.