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.
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.
"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"