They are similar but customer is more someone who buys things at shops and restaurants. Clients are more like people who pay for a service. Like a medical or business.

A client can be a "customer" , but it's definition when compared with customer is someone is offering a professional service. i.e. real estate agent working with a client. Their relationship is a professional one and is offering a service to lookout for their client's best interest. Customers are generally people who come to you mainly to buy products or services you supply. Clients buy your advice and solutions personalized to their particular needs.