A nut is a fruit consisting of a hard or tough nutshell protecting a kernel which is usually edible. In general usage and in a culinary sense, a wide variety of dry seeds are called nuts, but in a botanical context "nut" implies that the shell does not open to release the seed (indehiscent).
Chestnuts are both botanical and culinary nuts.
Nuts being sold in a market
Raw mixed nuts, sold as a snack food
Image: Almonds in shell, shell cracked open, shelled, blanched
Fruits are the mature ovary or ovaries of one or more flowers. They are found in three main anatomical categories: aggregate fruits, multiple fruits, and simple fruits.
Longitudinal section of a female flower of a squash plant (courgette), showing the ovary, ovules, pistil and petals
A segment of an orange that has been opened to show the pulp (juice vesicles) of the endocarp