Great question- thanks for asking!
Public Groups are groups that are accessible to all members in your organization. You DO NOT have to be part of a public group to view content or make posts.
If you want to stay up to date on all the great information in a Public Group simply click "+ Join" to add yourself to the group. Once you do this, all of the group updates will appear in your Group News Feed.
Private Groups are groups that are only accessible to those members who have joined. You HAVE to be a member of a Private Group to view content or makes posts.
If you want to access to a Private Group simply click "Request to Join" to add yourself to the group. Once approved by an admin, you will have access to all content in that group and all updates will appear in your Group News Feed.