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 :)