When it comes to mobile push notifications, your mobile app registers itself with the platform's (Apple or Google) push notification service and gets a token. It then has to send your server that token and your server can send pushes to it by talking to the platform's notifications endpoint and quoting that token.

These third-party push notification services are essentially overlays on top of the above, don't actually change the workflow at all (the above is the bare minimum you'd need for the functionality to work) but introduce extra complexity, moving parts, and more importantly, extra rent you now need to pay to them.