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.
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