Remix: is the reworking or adaptation of an existing work, generally efforts are focused on creating an alternate version of the original mix. Also in most cases specific samples are provided from the original mix's author to the remixer, usually isolated instruments, synths, bass lines, guitar riffs, percussions, kick, etc.
Mash-up: involves the combination of two or more works that may be very different from one another. Usually it's done by grabbing entire song fragments/samples from anywhere.
Both terms are mostly associated with studio work, this is the main difference in comparison to playing a versioned (or same as original) cover live. You can also record a cover, but you are playing in essence the original composition with or without modifications, whilst mash-ups and remixes tend to aim towards a new or alternate composition from the original.
Mash-up: involves the combination of two or more works that may be very different from one another. Usually it's done by grabbing entire song fragments/samples from anywhere.
Both terms are mostly associated with studio work, this is the main difference in comparison to playing a versioned (or same as original) cover live. You can also record a cover, but you are playing in essence the original composition with or without modifications, whilst mash-ups and remixes tend to aim towards a new or alternate composition from the original.