Client and customer are two words for individuals (or even groups or businesses) who pay for goods and services. Both nouns are sometimes used interchangeably, but they differ in their conventional definitions.
A client is someone who requests the services of a professional. For example, lawyers, plumbers, freelance writers, accountants, and web designers often work for clients.

A customer buys goods or services from a business (rather than an individual or group of professionals). 

So, in practice, client is usually reserved for someone who pays for highly specialized professional services, while customer describes someone who makes purchases at a store or other business, especially on a regular basis.