The difference is regional and only slightly different in defintion. In the South (Mississippi, Tennesse, Louisiana, Alabama etc) you will hear "you" used more often in sentences. You will also here "now" at the end of sentences and Y'all at the beginning of sentences. If I heard, "Don't you get mad at me" I might think a Southerner may have said it. While, "Don't you get mad at me" can be joking (imagine kids laughing together) - OR it can be a very serious threat by someone who is very angry (maybe carrying a weapon for example) and pushed to their emotional limit. (Imagine a police standoff with a kidnapper who says: "Don't you get mad at me!" Like : Don't you get mad at me (or I'm going to do something if you do) kind of thing. Don't get mad at me! < This is more of a request or wish by the asker. Not a sarcastic or funny comment or threat. It's more of a plea from a person who did something wrong. In pronunciation - Don't get mad at me. < very evenly spoken. Don't YOU get mad at me! < emphasis on the YOU which makes the feeling stronger.

Honesty, There's no difference at all. If there was one, I would say "'Don't you get mad at me" would be more specific.It means you don't want "the person" to be mad. While, "Don't get mad at me" would be less specific, because you could be talking about anyone. Also, "Don't you get mad at me" Is often said in a sarcastic or playful way, meaning they aren't really mad you're just teasing :P (I'm online right now and your questions are tempting me XD)

The "you" makes it very personal. In a nutshell.