I am trying to figure out how My Activity and Recent Activity differ (i.e. Menu items). From what I can see they look to contain the same series of posts.

My Activity = Activity you've created or is targeted at you.
Recent Activity = Everyone's Activity that you have access to

Is the Activity Tab in an individual profile the same as "My activity" menu item and contains just things they've engaged with to would it also include posts from connections?

Yes, but it only shows your connections activity to you while you are looking at it. Others viewing your profile do not see connections activity there. This is the same as Facebook in that when you are viewing it it's like viewing your Home stream.

I gather to control who sees which posts is done via the Privacy Plugin? Is what appears visible in the recent activity also based on what your connection's privacy setting are?

Yes, you can control access to activity using CB Privacy. Yes, inaccessible profiles should have their activity removed from the Recent Activity page.