More or less. It depends on the context as to which sounds natural, but customer is generally used more. Client is often used for interactions between businesses. For example, "the lawyer has many businesses as clients".

Professionals offering a service have clients - lawyers, designers, architects. These clients can be private individuals or other businesses, but they pay for a service which is professional, complex and often bespoke. A customer is anyone who purchases gods or services, but the service would be simply and not personalised, e.g. a mobile phone contract.