Fårö Island is in the central Baltic Sea, southeast of Sweden (just north of Gotland), and administrated by Sweden.

The Faroe Islands (or the Faeroes, as that’s another acceptable spelling, as used in the British “shipping forecast”) are in the Atlantic Ocean south of the Norwegian Sea, north-northwest of Scotland, and owned by Denmark.

Farø is a tiny island in Denmark proper, between Zealand and Falster.

Not at all. The Faroe Islands are between Iceland and Norway.

However, it seems they're almost named the exact same thing. The Faroe Islands means "The Sheep Islands" while Fårö just means "Sheep island"