They both mean tell me. Dígame is in the usted form, so you would use it with any adults with whom you are not very familiar. Dime is for children, peers, and adults with whom you have a friendly, more intimate relationship relationship.
Thanks... that was pretty basic, and it's actually what I thought it was. But I had asked a few people who's native language was Spanish and they would shrug and say "There really isn't a difference..." so I didn't know when to use which. Thanks!