A Push Notification is a message that pops up on a mobile device (similar to a text / SMS message). It will only reach users who have notifications for that app enabled (users can opt out of push notifications in the App settings).

An App badge is a small number that displays on the app icon, or on icons within the app itself. It alerts the user of unread notices, messages, updates, etc.

When a push notification is sent, the badge will only show on the App icon.

When a notice is published without a push notification, the badge will only show on the relevant icon within the App. For example: