Yes now, the two organizations have the same name. You can call it the mafia or cosa nostra. But at the beginning, the two were separate.
Al Capone was never a member of the mafia, he was simply an italian gangster. But after the birth of Cosa Nostra, he was a member of the Cosa Nostra.

Mafia was only for sicilians, even in the US.

But with time, and the use of the words by the news, cosa nostra and mafia became synonymous.

Also, with time, mafia became almost a synonym for organized crime, that why people use japanese mafia (yakuza), chinese mafia (triads and tongs), russian mafia (bratva).
Even in italy, some news call the ndrangheta the calabrese mafia and the camorra the naple mafia

You get the point, the use of the word mafia change with time

Even tough mafia should only be use for a type of organization from Sicily with only (or mostly) sicilian members