The term "client" stems from the client-server model that has been the foundation of the internet forever. The client requests information and the server gives it to them. "Client" by itself can refer to the device or the human user, while "client application" is talking about the computer software the client device or user is using.
Conversely, "server" is the computer hardware, and "server application" is the software. But, client is also used alone to refer to client application. Clear as mud, right?
So here's the deal, if someone asks you what "client" you use for mail, the answer might be "Outlook." They might also ask what "application" you use for mail, and the answer would be the same. The point is that the client is of no use without the information from the server. Another example of a client is a web browser, again, totally useless if you're not connected to the internet and able to retrieve data from a server.
In contrast, a "software application" (or application software, or application, or software, sigh) is a computer program or group of programs created for a human user, which processes data in certain ways based on its programming and user instructions. It's not a "client application" because it doesn't need a server. Once you have it on your device, you put in your own data and select the functions you want the program to perform on that data, like Excel.
To make things even more complicated, we now have "web applications," like Google Docs, which work the same, but actually live on servers rather than on clients. (Are you still with me?)
Even more recently, people are making a distinction between "application" and "app," which is obviously an abbreviation of the former. But in the tech world, an app has only one function, like a clock, calculator, or game, while an application has many functions. For example, Microsoft Suite, is an application (but so is Word).
Anyway, the purpose of this series on digital lingo is to make it easier for you to communicate with your tech team to get instructions, find a fix, or build something new. So, to summarize, Outlook is your mail client, which retrieves your mail from Shaw's servers. You add up your expenses on the application Excel, then check the game scores on your hockey app.