A cover usually refers to a
reinterpretation of the song. It takes the form of an acoustic version, uses
different arrangements or changes the lyrics. The remake is only a reproduction
of the song. It is sung by another artist but singed in the exactly way.
Normally the remake is singing the song
with the same base that the original one and with exactly the same lyrics and
rhyme. The cover is the same song but with any change. Usually the remakes
are recorded by the original artist but they can be sung by another.
Sometime a remake is made by the same
artist that did the first version. Maybe they need to fill up and album and
they just record again and old song to use it. On the other hand, a cover is never done by the same artist that performed the original song.
Nowadays these terms are not very distinguished
as people normally use them to talk about the same, so there is not much
difference. Technically they are different types of versions but almost anybody
does the separation between these terms.