'Don't worry' is normal English. It's a sentence in the imperative form, with the same grammatical form as 'Don't cry', 'Don't speak' or 'Don't laugh'. You're simply telling someone not to do something. 'No worries' is a fixed phrase. This means that you can't analyse the vocabulary or grammar of it, and you just have to accept it as it is. Like 'You're welcome', 'No worries' is a formula that has a social function, used as a polite response to 'Thank you'. It's an informal response that originated in Australia and is now becoming popular in Britain, too.

No worries is a more casual reply when someone wants to say to you "That's all right". For example "I am sorry I don't have that book for you" ."That's okay, no worries". It means "it doesn't matter" and is frequently used by Australians and British. "Don't worry" is more to do with the meaning of the verb "worry" - for example the doctor can say"Don't worry, your condition is not serious". It means "there is no need to be concerned, or worried about your illness."

no worries means no problem, don't worry means be not anxious

I tend to use "No worries" when someone thanks me (like saying "You're welcome") and "Don't worry" if someone thinks that they have been rude, or caused me trouble, or if they are concerned about something. "No worries" really is saying "It is fine", "I am fine", or "It is OK with me" so it sounds more gentle. "Don't worry" is more like an instruction to the other person, so sounds a little stronger, to me.

I'm not a native speaker, but as far as i know, both are correct and have the same meaning.