When it comes to paid advertising on Facebook, you cannot advertise a Group. You will need to refer people to an outside BLOG or website and send them back to your Facebook group from there. A Fan page though can easily be linked to Facebook advertising. Many times people will need to be invited to join a group (though some people will just join if they like what they see).

Make an informed Decision.

Before making a decision for your business on whether to create a Fan page or Group its important to review what you're objectives are for the page...what you want to achieve. A lot more can be done with a Fan Page in regards to adding functionality, where as a Group may be the answer if you don't need any of that, and its just a means to bring people together for one purpose.

One of the best ways to help determine the differences between a Fan Page and Group, is to actually go check some out. See how they work, how they look and how they function..and make an informed decision from there.