The one that we have in MySite, is it called Organization Browser?
And the one we add as a web part in publishing site page, is it called Organization Chart?
What's the difference between these two?
The Organizational Chart is a non browseable chart represented like a classic hierarchical tree. The Organizational Browser in MySite can be browsed if the Silverlight view is turned on and supported.
Read more: Use SharePoint to Build an Org Chart