In my opinion, I think the distinction has to do with the degree of the relationship between the buyer and seller. A customer is typically associated with a one-time transaction (buying milk at the grocery store) whereas a client has an ongoing relationship with the seller/service provider (a law firm representing a client in its legal matters). That is also why companies and professionals like to refer to their customers as clients, to foster a long-term relationship between the parties.

To me - there is quite an overlap, but typically a client pays you for services, and a customer pays you for goods. As Maddie says - someone who sells high-value goods will call their customers 'clients' to try to sound fancier. And conversely, 'customer' would technically include anyone who purchases from you.

There isn't much of a difference in terms of what the words actually mean - a client and a customer will both consume some sort of goods or pay for services to a company. A customer however, can be an ordinary person buying milk from the supermarket, and so, some prestigious businesses like to refer to their customers as clients. This is common in legal practices, real-estate practices, private businesses and banks, for example. So in a formal business letter, it may be best to refer to your customers as 'client', unless your boss instructs you otherwise, or the culture in your work is different.

I agree with Maddie in general. Perhaps if I quote the definition from the Oxford Dictionary and tell you how the terms were used in stricter times in London, you will have a greater understanding of these terms and then form your own judgment. The dictionary definition: "A person or organization using the services of a lawyer or other professional person or company". There was a time (up to the 1980s) in the world of business when only "professionals" such as lawyers, accountants, surveyors, engineers and architects could properly use the term "clients". "Professionals" were people who had to take special exams set by their chartered professional institutes to qualify to be partners in their firms. Bankers used "customers". Doctors and dentists used "patients". The world was much more formal, regimented and hierarchical then. Now, hairdressers call their customers "clients" because, it is argued, their profession is no less skilled than the law or architecture. Many other kinds of business practitioners and traders, including prostitutes, likewise call their customers "clients". So Sergio, you can form your own view on these terms and define your own principle of usage.