Pages and Groups both have their own individual functions, but one shines over the other for several different reasons (I'm talking about Pages, if you hadn't guessed!)
Pages Get Internal Promotion on Facebook.com
Facebook Page updates will show up on your fans' page feed when they "Like" it - Facebook Group updates will not. They are promoted on the right hand "suggestion" column when your friends join them - Facebook Groups don't get this treatment.
Pages Have More Options for Customization
Groups are basically chat rooms, there's no real "look and feel" to them. Their primary focus is for live discussions that can be continued at any time. Pages, however, act more like a blog with actual content pieces and the ability to add new tabs, static information, and the like.
Search Engine Visibility
When it comes to getting visibility off of Facebook: Pages can be seen by non-Facebook users, and can be crawled by search engines as any other static page would. Facebook Groups cannot! It's also important to mention that Facebook Pages can have a "vanity URL" (Facebook.com/MyPage), where as Groups do not have this feature.
The creator of a group is forever immortalized on the group itself, with your linked Facebook avatar, stating that you are the group's admin. There is no way to remove this, which is bad news for those who wish to remain anonymous. Pages, on the other hand, are completely anonymous - nobody will know that you own it, unless you make a public statement!
Groups can be really, really annoying
Know how you'll comment on a friend's status, and then get an email every time someone else comments under your comment? Groups do this. Just imagine this happening on a very large scale since the posts in Groups are more like chat posts. Groups that have many members = a lot of these posts, on a constant basis. This is a default setting, and many find it to be annoying.