Is there a difference between ダンス and 踊り? They are both defined as dance in the dictionary.
ダンス(dansu) is a Loanword from English. A word borrowed from English into Japanese (Anglicism, English Garaigo). Derived from the English word "Dance". ダンス is more used to refer foreign dance styles as Waltz, Hip-Hop, Ballet.
踊り(odori) is the traditional native japanese word with same meaning. An inflexion of the word "踊る(odoru)" meaning "to dance".
踊り is more general, referring also to the traditional Japanese dance. As 盆踊り(Bon-Odori) performed on お盆祭り(O-Bon Festival)(O-Bon Matsuri).
舞踊(buyou) and 舞踏(butou) are loanwords from Chinese, with same meaning. As in 日本舞踊(nihon butou) or the George R. R. Martin book "竜との舞踏"(Dance with dragons)(ryuutono butou) title.
舞い(mai) is also used, coming from the verb "舞う(mau)" also meaning "to dance".
"踊り" is also spelled "躍り" or "踴り".
Synonyms exists in any language. It's why 'Thesaurus' exists for the English language too.