Branches are for representing your departmental or hierarchical organization. If you are using extended enterprise, the branches can be directly associated to a sub domain. Branches are static but can be used  to enrol all users in the branch into a course, LP, etc, for reporting and for your PU permissions.

Groups work by adding users manually (static group) or by using the criteria options (automatic groups) to build groups of users based on the user profile data and/or course status (enrolled, completed, etc). groups can be used in many of the same ways as the branches but are not for organizing your users, nor do they fully control PU access. 

Hope this helps.

Branches allow us to provide Power Users controls over ‘their people’. We are a federation and allow Member Organizations (MOs) to have their own admins for their people (i.e. users in their branch). I manage over 50 Power Users allowing them to manage their users because of the user of branches.

Groups can be configured to be the same as a branch but also allows you to be a little more restrictive about the users in a branch. For instance, if you wanted to group together folks that are in a branch AND have their language settings to French, you can do that with the automatic settings. This allows us to serve up menu/pages/content to the French speaking people in one of our MOs vs another language. For our users in Belgium we have three languages and instead of moving people around in particular branches based upon their language setting, we use groups to manage that.

Hope this helps.