Transfer Authoritative Primary:
From the TPAM System Administration Guide:
"The 'Transfer Authoritative Primary' function is used when you want to switch the primary role from one cluster member to another. This should only be used when the original primary member is up and running."
The Transfer Authoritative Primary would be something that is planned while the current Primary is running.
Take Over as Authoritative Primary:
From the TPAM System Administration Guide:
"The 'Take over Authoritative Primary' function should be used when the primary has failed, is not going to be brought back up, and there are multiple replicas in the cluster."
The option to 'Take Over Authoritative Primary' would only be used if the Primary has failed unexpectedly and has no possibility of ever coming back online, and you have multiple replicas.