A group is a social group, where all agree to be members. Let's say you have a set of 8 friends who love baseball and you all love to chat about everything baseball. In real life you get together frequently to watch baseball on TV, go to ball games, etc. and primarily discuss baseball. That can be done on Facebook, but not yet on Google+. You want "shared circles" where the name of the circle is the same for all 8 baseball friends and their chit chat about baseball only stays within the group.
People naturally group around interests like this. And online, it is a great way to filter noise and get relevant information.
I have a number of online acquaintances around my interest in passwords. At the moment, email is the most useful way to exchange targeted password information with them. One of us just started a password reddit - but that's kind of public and subject to the usual reddit nonsense. I follow 2 of them on Twitter but find most of the information not relevant to me because they have many interests besides just passwords.
In a private group setting, things could get more in depth, and stay focused on the subject in which we all have an interest.
With Google+ just starting out (even at 10m users), there's already a feeling of exclusivity and community. Groups would definitely help keep that feeling going as Google+ goes public, though, as long as it's implemented right. With public circles, groups, tags, etc being advocated, I hope the simplicity of Google+ isn't lost.
That said, I'm liking Google+ so far. The circle model lets you operate in a way that is similar to real life, unlike Facebook. IMO, wall posts are about 75% of Facebook, and Google+ is already superior to what they are doing.