Customer is more general; it's anyone who buys something from you, be it a one-time purchase or a relationship of several years. A client is anyone (individual or firm) who buys from you on a frequent, often (but not necessarily) contractual basis. So if someone buys a screwdriver from you once, you can call that person a customer, but not a client. But if that same person buys screwdrivers from you every month, that person is a customer or a client.
In addition, certain fields (such as law and social work) tend to use the word clients whereas other fields (such as retail stores and suscription services like cable TV and newspapers) tend to use customers.