These are some very well-written but long-winded answers. Here is the difference as far as CodeIgniter is concerned:
URL - http://example.com/some/page.html
URI - /some/page.html
Put simply, URL is the full way to indentify any resource anywhere and can have different protocols like FTP, HTTP, SCP, etc.
URI is a resource on the current domain, so it needs less information to be found.
In every instance that CodeIgniter uses the word URL or URI this is the difference they are talking about, though in the grand-scheme of the web, it is not 100% correct.
Here is my simplification:
URN: unique resource name, i.e. "what" (eg urn:issn:1234-5678 ). This is meant to be unique .. as in no two different docs can have the same urn. A bit like "uuid"
URL: "where" to find it ( eg https://google.com/pub?issnid=1234-5678 .. or ftp://somesite.com/doc8.pdf )
URI: can be either a URN or a URL. This fuzzy definition is thanks to RFC 3986 produced by W3C and IETF.
The definition of URI has changed over the years, so it makes sense for most people to be confused. However, you can now take solace in the fact that you can refer to http://somesite.com/something as either a URL or URI ... an you will be right either way (at least fot the time being anyway...)