Disable-Mailbox : It will detach the mailbox(removed exchange attributes) from the Ad user objects and adds detached mailbox to the disconnected mailbox list
Eg: Disable-Mailbox username
Delete-Mailbox: It will delete Mailbox and Ad object and moved to Disconnected mailbox list. Disconnected mailbox will exists until Deletion settings set in the mailbox store. Any time we can go and reconnect to the disconnected the mailbox.
Eg. Remove-Mailbox username -Permanent
-Permanent option will permanently deletes the mailbox and with out moving to Disconnected mailbox list.