To clarify possible questions within the differences of a profile, page and a Facebook group I want to share the primary purpose for which of each are used.

Profile is a key part of Facebook and It is the most used by everyone. Each user has a personal profile, while the group is a place where, initially, Facebook allowed companies and groups have the ability to communicate.

The personal profile allows you to create a private space where you can share updates, photos, links, news, ask questions among others. All users of Facebook should have your personal profile to use other social networking features.

Groups provide an enclosed space for small groups of people to communicate about their common interests. Groups can be created by any user.

Pages allow the real organizations, businesses, celebrities and brands to communicate with your audience. The pages can only be created and managed by authorized representatives and officers of each organization, entity or company.

In simple words:

Profile – Each user has a profile for personal use.

Group – Center for common interests.

Page – Place of a brand, celebrity, organization, and / or company official on Facebook.

In more detail, these are the things you can do in each of them according to Facebook:

Profile

• Privacy: visibility of information published in a profile can be controlled by the privacy options.
• Public: Depending on how your privacy settings are set the public may
  send an invitation to create a
  connection. If you wish, you can limit invitations and who can see your public profile.
• Communication: The updates can be viewed by
  users with whom you have a connection. Any updates you make on other profiles or pages will be created
  under his name.

Pages

• Privacy: information of a page and its
  updates are public and generally
  accessible by most people on Facebook.
• Public: Any person
  or user can click the “like” on a page and create a connection to receive news and
  updates. There is no limit on how
  many people can click “like” on a page.
• Communication: Administrators of a page can
  share updates under the name
  of it. Updates are
  displayed in the “News feed”
  of the people who gave him “like”
  on the page. Administrators
  can also modify the
  tabs of the pages and review the statistics of activity and growth.

Groups

• Privacy: In addition to having an open group to the public,
  there are more privacy options available for groups. In secret
  and closed group makes
  its updates visible only to other group
  members.
• Public: Group members must be
  approved and added by other members.
  When a group reaches a certain size certain features are
  limited. The most useful groups tend to be the one
  created to share with some people we know.
• Communication: In groups, members are notified as their main option. Group
  members can participate in “chat,
  upload photos to albums shared document collaboration, and invite all members to group events.”