The most important difference between IMAP and POP3 first:
Emails received via POP3 are stored on the end device, while messages received via IMAP are preserved on a server, where the current state of the message (read, unread, replied, etc) is also stored centrally.

The most important advantage of IMAP:
You can access your e-mails from an infinite number of end devices and benefit from a constantly synchronized database. This is not possible with POP3.

POP3:

POP3 is a protocol for retrieving incoming e-mail messages. It was commonly used in the past when people only used a single email client program (on their home PC) to retrieve emails.

Advantages:

• No online connection required (only for sending/receiving).
• Retrieval of all or individual emails for local storage.

Disadvantages:

• Local storage at the querying email client.
• No central backup
• No synchronization with other devices

IMAP

IMAP is a protocol for retrieving incoming e-mail messages. It is currently preferred, mainly because of its synchronization capability across many different querying devices.

Advantages:

• Central storage at the mail server
• Central backup at the mail server
• Flexibility due to anytime access to any mailbox
• Independence from querying devices (query at any time via any device shows the same mailbox status)
• Up-to-dateness: always and everywhere the current version of the mailbox
• Synchronization of the mailbox to all querying mail clients
• Majority of clients store local copies of messages (like POP3)

Disadvantages:

• requires online connection.
• overflows (“mailbox full”) if emails are never deleted.

CONCLUSIONS:

For modern email systems, the use of IMAP is recommended over POP3 because most people use multiple email clients to retrieve mail and only with IMAP there is the same level of data everywhere.

See also:
Mailix Documentation: What is POP3?
Mailix Documentation: What is IMAP?