'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.