Hazelnut, the fruit of the hazel tree, is shaped like a chestnut, has a hard shell, and has a fat, white and round kernel. It is fragrant and contains a lot of oil. It tastes particularly delicious and has a lingering aftertaste, so it has become the most popular. One of the nut foods, it is called the "King of Nuts" and is called the "Four Big Nuts" together with almonds, walnuts and cashews. (Quoted from Baidu Encyclopedia)

The two are of course different. Chestnuts are generally larger than hazelnuts and have different shapes.