I've joined Facebook groups and pages right and left since I joined up a few months ago - without really knowing or caring about the differences. So the Howard Greenstein article on Mashable (great site name!!) was very helpful delineating between the two.
I've become a member of several groups on Facebook. None library related. I'm in Pre-Code Hollywood, and Slam Bang Theater, The Underground Garage and When I Was Your Age - Pluto Was a Planet, among others. In honor of 23 Things - I joined the Texas Library Association Members' group. I joined the cause: I Love Banned Books.
I was a member of the Kimball Art Museum group - but they changed to a page. I'm a fan of their page now, and get regular updates on what's going on w/ them. The same for the Modern. For 23 Things, I've become a fan of the Libraries page. This lovely page has pictures of some of the more famous and historic libraries all over the world. Some other pages that count me as a fan: Texas Daytrippers, Texas Native Plant Society, Dallas/Fort Worth Thunderstorms, various arts & music pages. Among others.
The Library should have a page (or pages, even) & invite everyone in our database of emails to become fans. That would be a free & extremely simple way to notify our patrons of upcoming programs, news, new materials... everything.