Usually the one doing the cover has to pay the original songwriter. That can be done by splitting the royalties or "buying" the rights to cover upfront, depending on the options the rightholders give you. For Spotify it doesn't matter, they pay exactly the same in theory.

In praxis Spotify often has special deals with big record labels, so covers technically make them more money since they only have to pay the standard cut to the artist.

Songwriting and recording/performance have separate royalties attached to them. You can transfer your rights (to labels, for example) and you can split them (one person wrote the melody and one the lyrics and agree to split) so it can get complicated. Obviously they often overlap and you could own both. Anyways, for a cover the songwriter should get some money for songwriter royalties but it doesn’t matter who previously performed the song, nothing for them.

Spotify also owns some of its own labels and artists usually give up their copyright when they sign with a label. What percentage of the royalties they get is basically a negotiation with the label at that point regardless of the song credits listed.