Well, botanically speaking: nuts are classified as fruits.

Though some nuts we call nuts in everyday language are actually seeds like Brazil nuts, whom we call nuts but who are seeds. The hazelnut is a nut, but a nut is fruit.

And berries are fruit, but there are different from other fruits.

A berry is a fruit that grows from the ovary of a single flower.

Whereas a nut is a fruit that grows a hard shell and a seed .

There is also berries that can be nuts like the strawberry as mentioned here.

The general term, fruit, in botany, refers to a seed -bearing structure in flowering plants.