This is for the front-end Fan Pages and Groups that users can make. A Fan Page can be Liked and a Group can be Joined.
We Like a page Page because normally a page Page would be a business, famous person, political figure, etc and so we would Like those sorts of things (well Well some would rather dislike the political figures).
We Join a group Group because normally a group Group is for things like clubs, self-help groupsteam, gathering etc where members would like to Join that sort of thing. Members normally Join groups to be a part of the group and participate in group things.
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If you make a Group, you can set it to be Public, Private or Secret in the permissions and you can't do that with a fan page.
A group Group has more privacy settings as a membership type of feature needs more privacy. The Fan And a Page would not involve more privacy because are normally public promotional areas.
Also, the Group will show a block to see the members of the Group, whereas, a Fan Page will show a block with followers that like it.
Group:
Page:
