How are Facebook pages different from Facebook groups?

Facebook answers that question in the Pages Help Center.  Their answer:

Pages can only be created to represent a real public figure, artist, brand or organization, and may only be created by an official representative of that entity. Groups can be created by any user and about any topic, as a space for users to share their opinions and interest in that subject. Pages can be customized with rich media and interactive applications to engage Page visitors. Applications can't be added to groups.

Pages are designed to allow Page admins to maintain a personal/professional distinction on Facebook, while groups are a part of your personal Facebook experience. If you're a group admin, your name will appear on that group, while Pages will never display their admins' names. Additionally, when you take actions on your group, such as posting on your group's wall, these actions will appear to come from you as an individual. However, if you post or take other actions on a Page you own, it will appear to come from the Page.

As long as a group is under 5000 members, group admins can send messages to the group members that will appear in their inboxes. If the group exceeds 5000, admins can't send messages to all members. Page admins can send updates to fans through the Page, and these updates will appear in the "Updates" section of fans' inboxes. There is no limit on how many fans you may send an update to, or how many total fans a Page can have. It's also possible to restrict access to a group, so that new members have to be approved, but access to a Page can only be restricted by certain ages and locations.