Modern applications often use both client-side and server-side routing in a "mixed" or "hybrid" way so it is quite hard to draw a line between these two techniques.
To better understand when and how to use server-side routing and client-side routing, you probably have to figure out what it happens when you have a large application that is used to manage a large manufactoring company (this does NOT happen very often in the real world. It is just a useful example).
In these cases, you will likely have different people (with different roles) who see different parts of this complex environment (different aspects or domains). For example, an engineer would see a file server with a lot of digital documents while the people working in the company canteen would see the menu to be prepared, the working schedule and the store. These are totally different application "domains" that requires totally different UIs so it makes sense to serve different SPAs to each type of user.
In this case, you could use server-side routing to serve a specific UI (SPA) to a specific user while you could use client-side routing to navigate inside this UI (and to load data). Think to these SPAs as "dashboards" or "control panels" devoted to specific "tasks" and used by specific types of "professionals".
For example, you could have a /myapp/engineering route for all of you engineers and a /myapp/canteen for all of your canteen staff. Each of these URLs would represent a specific domain and would serve a specific dashboard to a specific type of user. These URLs would be managed server-side.
Instead, you would use client-side routing to navigate inside each of these dashboard, loading data and re-configuring the UI as needed.
Of course, your app would probably also have a RESTful API used by your SPAs to fetch the data they need. The URLs belonging to the REST API must be managed server-side to perform their job (even if they are NOT associated to real HTML pages) and are invoked only by your SPAs "behind the scenes". They usually are kept in a separated "domain" like /myapp/api .
The same happens with static web page (like the "contacts" page and "about" page) that are usually kept in a /myapp/static folder (or "domain") and managed server-side (this folder or "domain" can be - and often is - hosted on a different server).
So, you should probably use server-side routing to separate applications domains from each other and client-side routing to navigate inside each domain.