After people tried (and failed) to initially connect with their favourite brands, bands & people, Facebook created Pages i.e. Nike, Coca-Cola, Apple & Lady Gaga all have a ‘Facebook Page’ associated to them; not a ‘Facebook Group’. If you’re an administrator of a Page, then, when you make posts, they are done on behalf of the brand i.e. Coca-Cola posted – "Let's catch up over a coke."
Pages, the go for Corporates
Pages are indexed by external search engines i.e. Google, as per a public Facebook profile while Groups on the other hand, are not. Access to a Page, can only be restricted by certain demographics such as age, location, etc.
Create a Facebook Page: www.facebook.com/pages/create.php
Groups work a little different, let start with an example here; I recently created a Facebook Group for high-school alumni 'Class of ____' . If I, or any of the administrators make a post, unlike the Facebook Page, the post comes from the administrator's personal profile, directly connecting the Group to the person.
Groups, the go for Personal groups/clubs
Groups are considered to be an extension of your personal actions on Facebook. Groups also offer more control with the levels of participation. Administrators of a Group can restrict access, so that new members must be approved, even make it an invite only so it’s more of a private club. Due to certain size limitations, only Groups <5K members can send mass emails to all members, which can be a very handy contact point.
Basically, you can (to a degree) anonymously manage a Page but not a Group. Pages allow you to act as a corporate entity, where as a Group does not.
Create a Facebook Group: www.facebook.com/groups/create.php
IF YOU ARE A CORPORATE LIKELY YOU NEED A FACEBOOK PAGE.
IF YOU ARE LOOKING FOR A PRIVATE SILOED CLUB, LIKELY YOU NEED A FACEBOOK GROUP.