"customer" is usually a less-customized or less-involved relationship, in which there is a short transaction. The focus is usually more on products than services.
"client" is generally something more involved, with a service focus. It's used for things that require more engagement, and personal interaction. Some kind of continuity of the relationship is implied...if you deal with someone on only one day and never see them again, usually that person would not be called a "client".
McDonald's has "customers" who come in and order hamburgers and leave. But if a graphic designer were hired by McDonald's to redesign their menus, this would probably involve interaction with a lot of members of management...sending drafts back and forth, etc. They would say McDonalds was one of their "clients".
Saul is a lawyer, and lawyers say they have "clients", not "customers".