Payment is the moment when you pay for something. For example when you buy something online you choose payment method i.e. credit card, paypal etc. If you go to a store and you buy something you can also pay by credit card or cash and what you do is the payment. Expense is a cost, the amount of money you have to spend to purchase something. For example if you buy a car for 10000$, 10000$ is your expense :)

expense means the cost required for something; the money and the other payment means the action

As the cost is the price you have something, (which costs) and expenditure is an outflow of money to buy something or whatever

According to an online dictionary, "expense" means "the cost required for something; the money spent on something" while "payment" means "the action or process of paying someone or something, or of being paid" or "an amount paid or payable". Usually you would use payment to talk about the amount you're paying someone, and expense to talk about how much it cost you.