Community pages are auto-populated by posts of people with similar interests based on what they like, follow, and click on, and the priority of posts or what’s featured cannot be controlled by a single user or admin.
On the other hand, Groups are created for a specific purpose and interest and is administered by the person or people who own the Group. Based on the settings, Groups can be open to the public, users can request to join or members can join based on invite-only.