It's one of the few remaining features needed to create fully functional web applications, on par with the functionality of a native app.
As an example, I refuse to install a native Facebook application because of privacy concerns, but a web version with push notification support could provide the same functionality.
- Changes to files in cloud/collaboration services
Basically, any time you would use a push notification on a mobile device, but when the "desktop application" is essentially just an electron wrapper of the web UI.
Redash sends you a web notification when a long running query has finished. Clicking on the notification opens Firefox to the right tab where you can see the results.
I guess you have some notifications on your phone, right? That's the point, notifications for things. If someone sends you a text then you need a notification so you actually notice it.
They're useful on desktop, for having things like Gmail open in the background, but they're not hugely useful on mobile. Usually if you want notifications you would already have the app installed.
The whole point is that if Apple would allow this, you wouldn't have to install the app. You could just use the website.
Forums are an example where the website is perfectly fine without an app, but I'd really like to be able to choose whether or not I get push notifications from them on my phone.
As an example, I refuse to install a native Facebook application because of privacy concerns, but a web version with push notification support could provide the same functionality.