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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 has more privacy settings as a membership type of feature needs more privacy. 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 Page will show a block with followers that like it.
Group:
Page:
