This is for the front-end Pages and Groups that users can make. A Page can be Liked and a Group can be Joined. 

We Like a Page because normally a Page would be a business, famous person, political figure, etc and so we would Like those sorts of things (Well some would rather dislike the political figures).

We Join a Group because normally a Group is for things like clubs, team, 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.

These have different settings allowed for them. For example:

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: