Was the term “sloth”, referring to one of the Seven Deadly Sins, named for the animal, or was the animal named for the sin?
I’m pretty sure the 7 deadly sins predate the discovery of the Americas (where sloths live), so it’s probably safe to say that the sins came first.
Regardless of how old the sins themselves might be, it is clear to me that the OP was not talking about the sin itself, but about the English word.
According to the Oxford English Dictionary “sloth” meaning “laziness” comes from Anglo-Saxon, and is etymologically related to the word “slow”. The New World animal was named after the abstract concept on account of its behavior.
The “seven deadlies” were codified by Pope Gregory in 590, so that also well predates the colonisation of the Americas.
As has been said, the name of the sin came first. In Spanish, sloths are called perezoso or oso perezoso, meaning “lazy one” or “lazy bear.” The name of the sin is pereza.
Sloths were first described by Gonzalo Fernández de Oviedo y Valdés in his Historia general y natural de las Indias in 1526. Rather bizarrely, he called it a periquito ligero or “light parakeet.” He says of it “The sloth is the stupidest animal that can be found in the world, and is so awkward and slow in movement that it would require a whole day to go fifty paces.” He kept one in his house, and because he never saw it eat he believed it fed on air.
The first (very strange) published illustration was by the French writer Andre Thevet in his Les Singularitez de la France Antarctique (Singularities of France Antarctique) in 1557. Thevet referred to the animal as Haut or Hauthi, evidently derived from a native name for the tree where it lived.
Linnaeus described both the two-toed and three-toed sloth in 1758 in the genus Bradypus, “slow-foot.”
In German, sloth is “Faultier”, which means “lazy animal”. The deadly sin though is “Trägheit”, of which one translation is sloth.
My go-to for any etymology questions is etymonline.
He notes that sloth, meaning laziness dates to late 12th century, while the animal dates to 1610, and is a translation of the Portuguese preguiça