Mbox is an email storage first implemented in 1974 for Fifth Edition Unix.
Maildir was developped in 1995 by Daniel Bernstein for his mail transfer agent (MTA) qmail, addressing technical issues inherent to the mbox format, especially solving file locking problems when new mail is being received.
The mail difference between the two is that mbox stores all of a user's emails into a single file, while Maildir stores each single email in a different file. All the major advantages of Maildir over mbox derive from this single fact.