App Center Push vs Notification Hub

I think its important to add the fact that App Center Push, Auth and Data are now being retired, despite Push being a fantastic offering IMO:

https://visualstudiomagazine.com/articles/2020/02/14/app-center-mbaas.aspx

"We are discontinuing efforts in the Auth, Data, and Push services and working to retire these preview services in App Center," said John Wargo, principal program manager for Visual Studio App Center. "With this change, we will focus App Center on delivering a world-class mobile and desktop DevOps experience. We will also work together with Azure teams to help migrate developers to the native Azure services, and ensure that Azure continues to be a great platform for your mobile apps."

There seems to be no full migration path at time of writing, but there is the hint this will be handled mostly for existing users -

https://learn.microsoft.com/en-us/appcenter/migration/push/

We don’t expect customers to manually migrate from App Center Push to Azure Notification Hubs; so before the service shutdown, we’ll provide detailed instructions for how to streamline your app’s migration from App Center Push to Azure Notification Hubs.

They don't have a reference from either to the other because they are two completely unrelated solutions (except for the fact they solve the same problem).

Both allow you to setup a cross-platform push notification infrastructure for your mobile app, but App Center is free, specifically built for iOS, Android and Windows notifications and requires less setup, but gives you a bit less control over how the notifications look. However it makes it easier to segment notifications based on analytics data. Notifications Hub is a Azure service, so you need a Azure subscription and you need to build a backend to make it work. It takes a more work but you have more complete control over the notifications.

The awesome James Montemagno made a nice summary article comparing the two options.