Question: I don't understand the difference between who can access the department messages versus just their direct messages.
Answer: You can configure who has access to any department message on the administration page, under "assignments".
Here you can see this user has the ability to send and respond to messages for the Career, Admissions, and Student Life departments.
All users that have "Send And Respond to Messages" toggled on will be able to have direct message conversations with students. If they also have any of the boxes checked under "Assignments", they can send and respond to messages for that department.