A remix is a literal re-mixing of the original recording. The masters of the original recording are re-arranged, some stuff is taken out and extra crap is put in. An good example I could give would be the White Zombie album Supersexy Swingin' Sounds. Every track was originally recorded by the band, but re-arranged by Rob Zombie.
A cover is a completely different thing. A cover is when a particular band records or performs their own version of a tune previously released and written by someone else. For example, my band used to do a cover of "My Own Summer" by The Deftones. The album Garage Inc. was an entire album of covers done by Metallica.
Metallica's covers of basically anything are usually better than the original. There's not one person in the world that can honestly say they think the original version of "Breadfan" is better than Metallica's version.
The only one that might be the exception to the rule is "Stone Cold Crazy".