Shared Mailbox - Add-MailboxPermission vs. "open additional mailbox" in Outlook itself

Again, the "older" methods also requires permissions to be granted, so the difference in behavior is mostly due to the difference in permissions. The "older" method is still working and supported, mostly used in cases when you only need to share a single folder (the generic case of sharing a Calendar or Contacts folder is exactly that). It's also preferred by organizations switching from Notes.

The Full Access method on the other hand allows you use many additional Outlook functionalities, especially when adding the mailbox as Additional Account. Automapping also has its uses, for example when you need access to the Archive mailbox. But overusing/over-relying on Autodiscover will cause you problems with bloated Outlook profiles, large OST files, etc.