Is there a difference between a cover and a remake of a song? Or do both terms mean the same thing?
It's a bit wishy-washy and the terms can be used interchangeably. However, I would say that in general:
A cover usually refers to a reinterpretation. This might take the form of creating an acoustic version, a different arrangement, or even just taking the lyrics and coming up with an entirely new melody.
A remake is an attempt at reproduction, playing the song the same way it was originally played. The remakers might put their own twist on it or they might get very technical and try to be exacting, but ultimately it's the same style as the original.
I think though that most of the time they are not distinguished. Wikipedia says that a cover is simply one band/artist's songs being performed by another. It also gives some interesting insight into its origin:
The term "cover" goes back decades when cover version originally described a rival version of a tune recorded to compete with the recently released (original) version. The Chicago Tribune described the term in 1952: "trade jargon meaning to record a tune that looks like a potential hit on someone else's label."
Cover typically refers to a new performance or recording of a previously recorded, commercially released song by someone other than the original artist or composer. The new performance or recording usually is very similar to the original (structure, progression, etc), the differences are normally in instrumentation. Some cover artists are very serious and careful about playing their covers as close as possible to the original, others inject something new, but in general both the original and the cover are very similar.
A remix (including this for completeness) is also a reinterpretation of the original song, but with more liberties. Changes in key, structure, and progression are very common. Sometimes a remixed song can be indistinguishable from the original.
In my experience, "remake" doesn't have a formal definition, even in this specific context. It can be a synonym of reinterpretation (a general term that includes cover and remix), or refer to an updated version of a song (1985 vs 2013), or as a cover synonym, from the top of my head.
As "remake" is used in general terms, it makes sense for it to refer to reinterpretations in general. As in both a remix and a cover are remakes.
In short, remake refers to redoing something, carving the song again. The details of the remake are what give it a more specific name, like cover or remix.
The term 'cover' in terms of commercial popular music arose to describe the origination of a particular song. Money was generated by popular songs when they were deemed suitable for commercial production/being produced as sheet music and when recorded for commercial transacting. The identification of a song with a particular orchestra, band, and later, singer meant a certain monetary security for songwriters and music producers/agents. 'Covers' were initiated by those who owned musical copyrights to ensure property usage/commercial security AND popular/cultural identification of the song would achieve a broad 'airing'. Creating, effectively licensing, another to commercially use the 'rights' of a song helped to ensure the maxim monetary return on a song.
Music producers, songwriters, film companies, etc., began in the 1930s to designate the covering of an original song (and it's corresponding performance) to ensure commercial viability over time, and not merely in the months following a release of a song as a record or when performed on radio. So, to 'cover' a song meant to re-present a song to the masses, and that of course came to mean artists/musicians/vocalists would 'reinterpret' the original. By the 1940s preferred artists such as Bing Crosby were often given the initial refusal of a particular song. (Though the practice did happen in the 1930s with Crosby and Russ Columbo and Fred Astaire, etc.) And then a secondary artist (an alternate artist) was extended the opportunity of recording the song after a designated period. The 'covering' of songs began to be so profitable that 'licensing' of material/songs became a standard working practice for record publishing and song production. Songs were performed then covered by bands, the male singer, then a female interpretation, etc.
With the arrival of the LP (long playing - album) and the solo vocalist as a star entity within music, it became necessary/profitable for songs to be broadly recorded. By then the term 'cover' was being used to refer to anyone who recorded a song not originally of their own repertoire. Today the term is used incorrectly... one cannot 'cover' a song by Frank Sinatra originally recorded, say, in 1947... that's more properly a 'rendition'... Just as Sinatra singing a Fred Astaire original 20 years after it appeared in an Astaire film wasn't covering the song; he was interpreting it. Just as Harry Connick Jr. and Micheal Buble aren't 'covering' Crosby or Ella Fitzgerald or Sinatra, they are interpreting songs from the "American Song Book" when they perform/record those classic popular songs/standards from the middle of the 20th Century today. IF, one wants to be historically accurate.
There's really no difference. According to Wikipedia:
In popular music, a cover version or cover song, or simply cover, is a new performance or recording of a previously recorded, commercially released song by someone other than the original artist or composer.
The article seems to use remake and cover interchangeably:
In R&B, remakes are common, often seen as tributes to the original artist. R&B artists such as Mary J. Blige have recorded several covers of legendary R&B artists like Aretha Franklin "(You Make Me Feel Like) A Natural Woman," Chaka Khan (Sweet Thing), and Rose Royce "I'm Going Down." In 1995 D'Angelo remade "Cruisin'" originally recorded by Smokey Robinson. K-Ci Hailey of Jodeci recorded a remake of "If You Think You're Lonely Now" by Bobby Womack. Lionel Richie and various contemporary artists recorded and released a remake of "We Are The World" for Haiti in February 2010. In 1994 Aaliyah recorded a cover of The Isley Brothers classic "At Your Best (You Are Love)" on her debut album Age Ain't Nothing But a Number. In 2001, Christina Aguilera, Mýa, Pink and Lil' Kim recorded a remake of "Lady Marmalade," originally recorded by Labelle, for the Moulin Rouge! soundtrack.
The links section provides a link to the Remake article, and while it's specifically targeted toward film, it seems to have basically the same definition as cover (emphasis mine):
In film or television, a remake is a motion picture based on a film and television series produced earlier. The term remake can refer to everything on the spectrum of reused material: both an allusion or a line-by-line change retake of a film. However, the term generally pertains to a new version of an old film. A reproduced television series could also be called a remake.