Largely a customer would be buying physical goods (things), a client would be paying for services (for the other person to do specialist tasks on their behalf).

A shop has customers, Amazon has customers but a lawyer would have clients.

No problem!

Also, note how I used different forms for "to buy". 

You would tend to say "buy" or "purchase" for a physical object but not for a service.

Historically you would always say "pay for" a service. For example you would not say "I want to buy a haircut" but in modern casual English it seems that this rule is becoming more relaxed. 

Correct examples:

"I want to buy that hat"

"I would like to purchase this book please"

"OK great, and how can I pay for that service"

"Hi, I would like some English lessons please." ... "Ok, excellent, will I need to pay before or after"