Maildir will have each individual e-mail message stored as a file. In each mail account's folder on your file system you will see three directory: new, cur, and tmp.
As new message arrive at the mail account, the files for those messages are stored in the new folder.
Mailbox works on a single file structure. All e-mail messages are stored in a single file on the account, usually inbox.
In terms of IMAP folders, on mailbox these folders are actually new files like inbox but would be named the folder's name and again would contain all of the e-mail messages in that "folder".
With Maildir new "folders" are subfolders on the mail account, which again have the new, cur, and tmp directories beneath them.
The advantage with maildir is in the case of a large mail account with many megabytes of messages. Instead of loading up the entire inbox file or mailbox file, each individual message can be opened separately because each individual message is a file on the file system.