Fanpages and groups have their distinct roles on the social network. Firstly, a fanpage has fans, whereas a group has members. You can have a fanpage about a product, brand, celebrity, website or whatever and anyone can come and like the page and share their comments as well. But, a group would be more appropriate if you want to build a relationship with it’s members.
A fanpage can be on it’s own. In fact, you don’t need to have a Facebook profile to create a page. On the other hand, a group is linked to an admin. The admin has control on the group, he can add or remove members, he can delete posts in the group, he can also add others as admin too. To join a private group, the person needs to make a request and the admin chooses to accept or reject the request. Existing members in the group can also add members to it.
Of course, both have similarities as well, such as the ability to advertise upcoming events and posting on their wall.
I have recently created a fanpage for Zuberr.com, I hope you like it and become a fan.