Facebook Pages are great in the way that users know "who's the boss". If it's a business page, then you just post a new product/service and your users start sharing and discussing it. You can permit anyone from posting on your wall, so basically it becomes a blog-like place: update status, post news, make new custom tabs (this is the fun part).
Groups are better for those who share common interest and are EQUAL. Photograpy for example, whoever takes an amazing picture uploads it to the group, and users start discussing. Groups have admins who can control the access of users. It can be an open Group or a closed one.