A wallet is for keeping money in, especially notes/bills. It is generally flat, rectangular, around 10 cm long, and opens like a book.
The meaning of 'purse' depends on whether you are using British or American English. In British English, it is the female equivalent of a wallet. This is what women keep their money in. A purse is often smaller and more rounded than a wallet, and you open it at the top with either a zip or a metal clasp. It may also have a couple of compartments and pockets. A woman would carry this inside a larger bag, usually known as handbag.
In American English, 'purse' refers to this larger bag, also sometimes called a pocket book.