There is no much difference between a consumer and a client. The businesses which offer services rather than products call their customers as clients. Both consumer and client are the end users who actually use the product.
When it comes to the customer, a consumer may be a customer but not every customer is a consumer. To explain it further, every customer need not be the consumer who actually uses the product. He may buy the product to pass on to the actual consumer or he may use it for himself.