Facebook Pages are more of a front for businesses, designed for having only a few individuals as administrators and publishers. Whereas Facebook Groups is like a forum where people with common interests, hobby or background can socialize and interact.
The basic functionality of a Group is discussing, sharing, planning and collaborating. This is what you can do from the Facebook app or from the fb.com page from your browser. However, with Facebook Group app you can move between different groups, join new ones and manage notifications and discover more groups that appeal to your interest.
According to Facebook, this app is designed for those people who use Facebook Groups more frequently than their timeline. It provides them a smoother and faster experience. However, worth noting is that unlike the Messenger service which has been ripped from the main Facebook as a separate app; you will still be able to access Facebook Groups through the main app.
To use it you need the same Facebook token to login. It means that if you already have the Facebook app installed then it would ask you if you would like to login as the same user or otherwise. When you launch the application you will see all the Groups at one place with the more frequently used Groups at the top. You can create a new Group with just a click of a button. However, not everything is similar to the desktop Group version or the Facebook main app. There is a new tab in the Groups all called Discover. It suggests you more groups to join according to your interests, the Groups you friends have joined and on where you live.