One email address has two Internet Accounts I somehow have multiple accounts for the same email address - they both receive messages sent to that address but one (the POP) has hundreds of emails in the Inbox and Sent, going back years and the other (the IMAP) has four in, going back a week, and none out.

In Mail>Preferences>Accounts they both appear, one with a grey IMAP beneath it and one with a grey POP.

In Settings>Internet Accounts they also both appear, this time with a grey Mail and a grey Notes beneath them.

Is there any reason why I should want this setup, and what is the best way to rationalise it (which I would like to do by removing the POP, but without losing emails)?
You don't have two email accounts.  You just have two email protocols set up.

From Wikipedia:


  Virtually all modern e-mail clients and servers support IMAP. IMAP and
  the earlier POP3 (Post Office Protocol) are the two most prevalent
  standard protocols for email retrieval, with many webmail service
  providers such as Gmail, Outlook.com and Yahoo! Mail also providing
  support for either IMAP or POP3.


The major differences between POP and IMAP is that POP supports the "download and delete" methodology and IMAP "synchronizes" its content between the server and client thus allowing you to have multiple email clients updated.

With the setup you have, the POP protocol negates any advantage you would have gained with IMAP since it downloads and deletes the message meaning there is no longer anything for IMAP to sync.

So, which one to use?

If you want sync'd emails across devices, keep IMAP.  If you have only one device and you want to save your emails locally, use POP.

You can remove your POP email protocol by deleting it, but if you want to save your messages, you can create a "Saved Message Folder" under the IMAP account and move everything from POP there.  Once copied over, delete the POP account.