In fact, the common forms are 'You need not worry,' and 'You need to not worry.' The first piece of advice is something that would be given as a piece of advice about something specific - for example, if you were to visit the doctor you might be told this if the doctor decided that you only have a minor problem. The second piece of advice would be general advice given to somebody who has spent too much time worrying.

The second one is more forceful it means you have been spending a lot of time worrying and it has become unhealthy whereas the first one is just an off hand comment

The second one is someone is giving you their opinion or advice the first one is more general.