Arabian Nights vs One Thousand and One Nights

Arabian Nights and One Thousand and One Nights both are nouns.

Word	Noun	Adjective	Verb	Adverb
Arabian Nights	Yes	No	No	No
One Thousand and One Nights	Yes	No	No	No

As nouns, Arabian Nights and One Thousand and One Nights are synonyms defined as:

Arabian Nights and One Thousand and One Nights: a collection of folktales in Arabic dating from the 10th century

Other synonyms of Arabian Nights include Arabian Nights' Entertainment, Thousand and One Nights.

Arabian Nights (noun)	One Thousand and One Nights (noun)
a collection of folktales in Arabic dating from the 10th century	a collection of folktales in Arabic dating from the 10th century