Sorry for the slow response. I didn't have a quick answer for you at the time, but I'll take a stab at it now.
First of all, you are not the only one that is confused by the terms 'display mode' and 'view mode.' https://github.com/backdrop/backdrop-issues/issues/2342
The most common way that users use different display modes is when using views. I can display a view of my "post" nodes using any custom display mode.
Here is an example of me using a display mode I named "custom" (I should have thought of something better) instead of the default which would have been "teaser":
If you are able and interested in creating custom templates, you can create custom templates for any display mode, just by naming the template correctly.
If I create a custom template for my custom display mode, views will use that custom template including whatever custom mark-up I have included.
Some other display modules might ask you which type of display you want to use for your nodes and you can choose between 'default', 'teaser', or whichever 'custom' display mode you have created.
I know there are many situations where you might have this choice, but I'm not thinking of them right now. Certainly, in your custom code you might want to call or use a custom display for nodes.
Views is by far the most common use case, I think.