A customer is someone is someone who purchases goods or services that require very little attention from staff of the company. Supermarkets, shopping malls, automotive service shops etc will have customers.
A client is someone who receives a more personalized and detailed attention from the staff. Usually, it will be law firms, financial institutions, doctor's offices, therapists, even spiritualists, and car salesmen.
Sometimes, the line can be a little blurred between customers and clients. A shopping mall that sells brand name clothing will have customers, but a high end store within that mall might have dedicated sales persons that refer to their customers as clients. Sometimes, the moment attention is paid to a customer automatically makes that person a client. In these cases, I feel, it is done to make the customer feel a little more important.