To show the difference between intertextuality and remix, let us consider the following example.
The image shows Obama conversing with Putin over the phone, Obama asking what Putin will have for Thanksgiving dinner, the response being "Turkey".
One will fail to understand the pun/joke unless already acquainted with the context. And that is the recent (25 November 2015) downing of a Russian bomber by Turkish figure jets. Reports say that the Turkish shot the Russian airplane for allegedly entering Turkish airspace for as short as 10 seconds, which angered Russian President Putin who threatened a swift and harsh response.
The details that inform the meaning of the illustration are of intertextual nature; the text of the image at hand is only meaningful through the texts that are implicitly contained in it: the downing of the plane, Russian-Turkish relations, Russian-American relations, as well as the linguistic pun in "turkey"/"Turkey".
Remix comes into play in constructing the image itself. Two separate photographs of Obama and Putin talking on the phone, litte likely to each other (in these particular pictures), have been combined into one image, captions inserted in each, that produce a conversation between the two presidents, one that has never taken place.
Remix is how the work is made, it is the creative use of found material. Intertextuality is how the meaning of the work is created, it is the intertwining of texts that inform the comprehension of a given text.