They mean the exact opposite. The first implies that there is nothing that you are worried about, even if it were important. The second says that you are worried, but that the thing that is causing you to worry is really not important at all - nothing.

Is the same.

We don't use the first expressions as it is without other words so there is no point in comparing them. We say 'you don't have anything to worry about' and 'you've got/you have nothing to worry about' and also other variations such as 'you don't have to worry about a thing'. We would use your first one with the 'are you?' tag question at the end and the second we would use 'worrying' instead.

"You're not worried about anything." is a question as and can be used like : " you don't look good today, Your're not worried about anything are you? "You're worried about nothing" is a statement to reasure someone as in : "Hello Mr Jones, we have had you results back and I can tell you that you are worried about nothing , everything is fine.