How do Announcements differ from Activity Feed?

The Activity Feed is brief, central overview of your course.

The intent of the activity feed is is centered around those three philosophies: to simplify common teacher workflows, enable tools teachers love, and to help make the LMS familiar and friendly.  It does not replace the announcements tool.  The two tools naturally converge some functionality (for example in activity feed you can now create a message post w/o enabling comments) which is like a regular announcement.  When you have release conditions (release an announcement to a specific section or group), you rely heavily on replacement strings (welcome {firstName}) or you use course export/import of announcements year over year, use Announcements.  Otherwise, consider adopting activity feed as the way to share announcements, reminders, and to-dos within a course.