I would like to know what is the differences between the two types of groups.
appCore: Users --> Group management --> (able to assign users.)
appLms: My Courses --> any courses (Enter) + Teacher Area --> Group management --> Groups
Groups in the administration area are a way to organize users across the LMS. In addition to having nodes you can also have groups. And you can easily assign everything (courses, user areas in the LMS, catalogs.....) to groups as well as to nodes.
Groups in the teacher area are completely separate from the ones in the administration area. When you are in a course, you might want to separate students into smaller groups. The groups that you can create there can only be filled with users who are already subscribed to the course.