Every NSF file and template has what is called a replica ID. This ID is used to determine if two databases on different machines are capable of replicating with each other.
When you create a "New Copy", it creates a copy of the same database but changes the replica ID. So it becomes disconnected from the original database.
The option "Make Available Offline..." is the same as "New Replica" created on your local machine. The setting only becomes active if there is no local replica of the database you selected. It also shows different replica settings (smaller subset) so that non-power users don't get lost in all the settings of creating a new replica.