Customer is the general word to refer to someone who buys something from a shop. Client is used to refer to someone who pays for a service from a professional person or company. Patron is used to refer to someone who eats or drinks in a restaurant, bar, pub, etc.
We receive many more customers during Christmas. The customer is always right. A customer has just entered the store and bought the last camera that we had.
That lawyer has a lot of clients. The director had a meeting with a new client. The new hairdresser has already many clients.
Patrons are asked to refrain from smoking. We offer facilities for disabled patrons. Patrons are welcome to leave their comments on our guest book.