Atlassian Groups can only be created and populated by Jira Administrators. These are used to provide access to Atlassian licenses and are populated from AD. Atlassian licenses are granted by adding people to the appropriate license group on the AD side.
Atlassian Teams is a newer feature where Atlassian users can create a team and populate the members. A team dashboard is automatically created when you create a team. This dashboard will show tasks that are assigned to people that are added to your team and gives you a page to share information and Kudos. Warning that these teams CANNOT be used for managing access to licenses, permissions, assignment, or notifications. They can still be useful for team building and they will allow people in to use the @<team> mention to mention all the members of that team in a comment.
Assets LTS Teams are specifically being used for team assignments in the LTS Service Management project. These teams are managed by the appropriate team leads.