I think the difference is in the context, methods, tools, and limitations.
1) "Cover" is often used in a singer/band context, where the artist performs a song from another artist, most commonly associated with pop/rock. "Remix" is used in a production context, where the reconstruction is made using DAWs, effects, synths, edits, etc, most commonly associated with electronic music.
2) In general, a cover remains similar to the original in many ways. Melodies, structure, harmony, lyrics, etc, most elements will remain the same. If there are any changes, they'll tend to be in the arrangement. A remix, in the other hand, can drastically change the source work: different harmony, structure, add/remove/change lyrics, etc. The remix can hold no resemblance to the original, while the cover tends to try to resemble the original.
3) The remix uses material from the original (recordings, sequences, synth patches, etc). The cover doesn't (or tends not to).
4) Because of the above, generally, a cover implies performance and can be done without sound edit tools. A remix implies sound edit tools, and can be done without performers.
5) There's also difference in how the work is presented. Cover performers tend to replace the original author in the work's presentation: Morris Albert - Feelings, The Offspring - Feelings. Remix authors tend to present the work with the original and remix authors included, separately: Nathan Fake - The Sky was Pink, Nathan Fake - The Sky was Pink (James Holden remix).
Lets take Corona - The Rhythm of the Night as an example. The cover by Bastille (which is also a mashup) is clearly a cover? No original sample has been used in this one.
"No original sample" can mean different things. No sample from the original track? It doesn't fit the remix description. No original sample as in no originally created sample as in all samples are from the source track, then it fits the remix description. If structurally it resembles the original (harmony, melody, lyrics, rhythm, etc), it is on the cover area.
But there are remixes (remakes?) where they only keep the lead singers voice, and everything else is new: Simon from Deep Divas - The Rhythm of the Night. Is this a cover or a remix? And this one actually has remixes of itself too.
It depends. How was the structure and harmony changed? Do they try to resemble the original or are they trying to do their own thing? If only the voice is preserved, if really "everything else is new" as in harmony, melody, key, structure and everything else was changed, then it hardly fits the cover description and is more on the remix grounds.
Reinterpretations can be both covers and remixes (and neither), at least as defined by this answer.