Delegate access must be used to grant one permission to act on behalf of another person. It is a more generalized delegation of permissions over files and folders within one’s Outlook account vs. a detailed one, which is where folder permissions come in.
When you add someone as a delegate, you only have a limited amount of permission sets to choose from for your folders:
For example, if you need to grant permissions to your inbox and calendar alike, then you probably want to grant delegate access to the user with the desired permissions. Folder permissions, however, focus solely on that individual folder and do not require access to other mailbox files or folders to work. They also allow for more fine tuning of its permissions than what delegate access can grant.
If you want someone to have basic calendar permissions, such as being able to view and do basic edits to the calendar, then these permissions may be granted to that individual’s account and the calendar shared from the Outlook Calendars section. If you want another individual to be able to process your meeting requests and responses, then you need to use the delegates dialog as well.
For a detailed understanding of Delegate Access and Folder permissions, see here.