Folder Sharing enables another person to access one of your folders, perhaps while you are on vacation. However, it does not include permissions for one to act on behalf of the other. For example, a person who can access your folders cannot reply to email messages or respond to meeting requests for you.

Delegate Access is an Outlook feature that allows one person to act on behalf of another Outlook user. The most common scenario in which this feature is used is a manager and his or her assistant. The assistant might be responsible for maintaining the manager's schedule, including creating and responding to meeting requests. Some assistants might also monitor a manager's Inbox.

For further information regarding delegate/mailbox permissions, please refer to Microsoft's article here.