My favourite example to give for this distinction is 'news' vs 'nntp'.
An 'nntp' URL specifies a specific message in a specific group on a specific server, hence it always has an authority part. It's also indicating to use the NNTP protocol to retrieve the message.
A 'news' URI, on the other hand, identifies a specific message by its message ID, and leaves it to the user agent to choose a protocol and server.
(The distinction isn't quite that clear cut, as 'news' URIs can specify a server, making them more like URLs, but I think it conveys the difference.)
An 'nntp' URL specifies a specific message in a specific group on a specific server, hence it always has an authority part. It's also indicating to use the NNTP protocol to retrieve the message.
A 'news' URI, on the other hand, identifies a specific message by its message ID, and leaves it to the user agent to choose a protocol and server.
(The distinction isn't quite that clear cut, as 'news' URIs can specify a server, making them more like URLs, but I think it conveys the difference.)