I often hear the words hang on and hold on, especially on TV. People use them when they want someone to wait for something. What's the difference between them?
In the provided context, they are very similar.
Hold on can mean the same in general conversation:
While the end result is the same, when used as part of telephonic conversations, hold on often refers more to being placed 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.