Is "Happy New Year!" more commonly translated as "¡Feliz Año Nuevo!" or "¡Próspero Año Nuevo!"? Are the two basically synonyms, or is there a difference between the two?
I'd say "Próspero Año Nuevo" is more formal and not really use in speech. You'll find it in Xmas cards, etc.
"Feliz Año Nuevo" is used but for a more informal form you can just say "Feliz Año". "Feliz Año" is what you hear everywhere these days in Spain.
I would say
It's very common to say