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.
Both are metaphors... in the context about waiting a short time.
Hanging on the line, like a fish, or a person at the end of a phone line. Keeping some one hanging so you can think/rethink/restate your position. Hang on a tick while I look up the number - I'm not going anywhere. And I keep muttering, okay P, Pa, Potter, Powers - got it, 5367 4122.
Holding, keeping the same position, not going away - even though I am for a moment. Hold on a tick while I get him - I'm going to be gone for a minute maybe.
So I feel hold is likely to mean a more significant pause or interruption (either longer or more removed), rather than just a filler for thinking or looking something up.
If you're saying to wait a minute, either 'hold on' or 'hang on' will do.
But if you're talking about literally grasping something with your hands, you usually would say 'hold' and not 'hang.' For instance, you would say, "Please hold on to my plate for me." You wouldn't say, "Hang on my plate for me."
'Hang on' carries the connotation of holding onto something you're falling off of, like when you hang on to the edge of the cliff. 'Hold on' carries more of a connotation of lifting or grasping something.
But as I said, most people use them interchangeably.