"Customer" is way better for this case! By saying "Client" you're showing that there is a deeper relationship between the restaurant and the person who visited. I hope it helps Aga!

"Customer" is correct. Actually, I think "diner" is the most suitable word choice here, although "customer" is fine. "Client" is wrong. In the United States, "client" is used only in a professional relationship, and only for certain professions. Lawyers have clients. Accountants have clients. Social workers have clients. Oddly enough, though, the set of all of a business's customers can be called the business's "clientele."

Customer is the right word. You provide products to a customer, and services to a client. A restaurant provides food, which is a product. Tim's comment is correct, that you might use 'client' as an exaggeration, or as a form of pretentiousness. In other words, a restaurant that thinks that the 'dining experience' they provide is more important than the actual food, could say that.

Customer

Definitely customer unless you were trying to be ironic or exaggerating the customer relationship to be more than it really was.