UPDATE: As of 2021, ArcGIS content migration (such as items, groups, and users) from one organization to another is now transitioned from Admin Tools for ArcGIS to Backup My Org.
Copy
This tool creates a new item as a pointer to the original source, which is useful for multiple users that are all working on the same content.
Clone
This tool creates brand new items, with its hosted dependencies configured automatically, and no longer pointing to the original data source it was cloned from. This is a very useful tool for moving content into a new organization. The Clone tool is available in Backup My Org.