Disclosure: I work at Moengage.

I feel web push notifications are a natural extension to the push notification on mobile. Increasingly, almost all services are available as web apps and we having to download lesser and lesser apps onto our machines. Think of Slack (web version) or Skype for Web, these are the kind of services which need to interrupt you to notify you of a message or call.

Another use case is in sports. Enthusiasts diligently sign up for score updates and want to be aware of latest happenings in a match all the time, if they can't see the live telecast.

Web Push is also a significantly cost-effective channel compared to SMS or email with high click-through-rates. It is better performing (for better or for worse) in terms of engaging a user.

I feel we are moving towards a web based ecosystem and installing apps will be a thing of the past and web push is one of the many things that will help bridge the transition. (Think of PWAs).

The bad of web push is the same as bad of push on Android devices (where asking permission is not required for push notifications). End of the day, it works for marketers and they have more and more incentive to urge a user to accept their push notifications.