As a newbie in product management, these terms will look and sound as though the same. But they are very far from being the same. Because as a product manager, the triage in which you will operate in can be either. A C2C, B2C or an Internal product operation. Just as these set differ, so does the term client, user and customer differ.
A user is a client and a customer. A client is a paying user and is not a customer, a customer is a paying user and is not your client. They look familiar but they are different. Mostly anytime a user is referred to as a client, it is usually because this user spawned from a B2B relationship. For example, Andela building a product for maybe product X. Product X in this case is the Client. They are not just users, but they were users before anything. In a nutshell, it is just not professional referring to a business partner as a customer. Even though that is what they are. It is however paramount to label these users differently for easy management and prioritized communication.
Meanwhile, a customer is a paying user in a B2C enterprise. For example, the user of Spotify paying monthly for access to music. This makes the user a customer. But Spotify also has a free plan. Using this free plan without payment automatically categorizes you as a User.
Summarily, a user is an active person who has adopted a product and is actively in use of the product periodically. While a client is a user who pays for products that operate in B2B (Business to Business) like Andela to product X. A customer is a paying user of an open product for uninterrupted use, like Spotify. SaaS falls into both categories.
A simple way to classify your users is to understand what you are offering. B2B Paid Users = Clients, B2C paying users = Customers, Non-Paying Users = Customers. Every user must be treated specially.
I pivoted to product management from digital marketing, and it has been a mission for me to use my little blog to simplify my knowledge for people who have a little challenge understanding these concepts.