NEEDN'T OR DON'T NEED TO.

This is a difficult question because most of the times we can use both of them.

1.
Both these forms are used to talk about past events, but there is sometimes a difference in use. When we say that someone needn't have done something, it means that they did it, but it was not necessary. Didn't need to is also sometimes used in this way:

You needn't have washed the dishes. I would've put them in the dishwasher.

You didn't need to wash the dishes. I would've put them in the dishwasher.

But we also use didn't need to to say that something was not necessary under circumstances where it was not done:

The sun came out so we didn't need to take any rainwear on the trip.
2.

There is also a difference in use when these verbs are used to describe present situations. We can use both needn't and don't need to to give permission to someone not to do something in the immediate future.

You don't need to water the garden this evening. It's going to rain tonight.

You needn't water the garden this evening. It's going to rain tonight.

3.

However, when we are talking about general necessity, we normally use don't need to:

You don't need to pay for medical care in National Health Service hospitals.

4.

NEED can either act as a modal verb or as an ordinary verb. When it acts as a modal auxiliary verb it is nearly always used in negative sentences, as the above examples illustrate.

Para resumir: (en Español)

-Usamos Needn't y don't need to para dar permiso en el futuro inmediato.
- Pero sólo usamos Don't need to cuando hablamos de una necesidad general, no personal, algo común a todos.
- Usamos Needn't o  Didn't need to en el pasado para expresar la no necesidad de haber hecho algo en el pasado, pero que sí hicimos.
-Pero sólo usamos Didn't need to en el pasado para expresar que no tuvimos la necesidad de hacer algo en el pasado finalmente.