I've been told that the difference between the two has to do with the timing of the recording.
To my understanding, a cover is a recording that is released by another artist at or near the same time that the original version was recorded. Originally, a cover was recorded in an attempt to cash in on a song's popularity, though this may no longer be the case as the industry has changed much since the term was first used.
Contrarily, a remake is a song that is recorded by another artist long after the original has been released. So, while the Iron and Wine version of "Such Great Heights" would be considered a cover, The Dixie Chicks version of "Landslide" would be considered a remake.
Of course, today many use the term "cover" to describe any newer version of a previously recorded song, and that bothers some who maintain that there is a difference between a cover and a remake. I'm sure remy will be able to elaborate more on the subject, but there you have it.