In OS X Mail, what's the difference in checking new e-mail between "Automatically" and "Every X minutes"?
Unfortunately Apple Support is a little vague on "Check for new messages" in Mail (El Capitan): General preferences.


  When Mail gets new messages from the mail server. By default, the option is set to Automatically (Mail varies how often it gets messages, based on whether your Mac is plugged into an electrical outlet).
  
  An IMAP account may get messages more frequently if the mail server supports the IDLE command; make sure the option to use it is selected in the account’s Advanced pane in Mail preferences.
  
  If you set the option to Manually, click the Get Mail button  in the Mail toolbar to check for messages. (Whenever you click an IMAP or Exchange mailbox, it syncs with the server, and shows new messages.)


It's not at all specific about "(Mail varies how often it gets messages, based on whether your Mac is plugged into an electrical outlet)" and how often it is between the two states.

At least with an IMAP account the "Use IDLE command if the server supports it" from Mail (El Capitan): Advanced accounts preferences says:


  The mail server pushes messages to your Inbox. When this option is not selected, Mail checks for new messages based on the setting in the General pane in Mail preferences.
  
  This option is available only for IMAP accounts.


So the difference is when set to Automatically it will vary on different factors while when setting it to "Every x minutes" you know it's going to check however often "x" is.