They are close, but not quite the same. If I go to a store to buy something and may or may not return, then I am a customer. If I have a long-term relationship with the store to by something on a regular basis, then I am a client. I go to an office supply store one time to buy a ream of paper. I may or may not be back. I am a customer. I got to the same supply store on a regular basis and receive a special discount because I buy many things from the store. Then I am a client. This would especially be true if I had some written contract with them.

In short, service with clientele denotes a contractual relationship whereas customer service implies no formal relationship between two parties, although it can exist.