Both expressions are correct grammatically and mean the same thing. The main difference is between their level of formality.
“Don’t you worry” is much more informal and is not something you would say in a formal situation. For example, in a job interview, it is not appropriate to say, “Don’t you worry. I’m a fast learner.” Instead, say, “Don’t worry. I’m a fast learner.”
On the other hand, “Don’t you worry” is often used between friends or family members:
“Don’t you worry, Marie. I’ll give you a ring as soon as I get there.”
“Oh, your dolly’s leg is broken. Don’t you worry, Dad will fix it for you.”
While “Don’t you worry” may sound like a nice way to reassure someone, in reality, it is often used in a condescending, patronizing manner. It makes the other person feel like a child who is told that he or she is just being silly and should stop worrying when there is nothing to worry about. So, if you choose to say it, please make sure you say it in a friendly tone.
*condescending, patronizing = showing that you think you are better/more intelligent than the person you are talking to
