A quote is when you repeat something that was said by someone, usually a famous, funny or eloquent person. Depending on the content of the quote, it can be used in any of those situations. An idiom is a phrase that sounds like something different than what it means. It creates an image of one thing that usually doesn't have much to do with the actual meaning. The most famous example is "raining cats and dogs". It means raining hard, it has nothing to do with kittens and puppies falling from the sky. I think that it would be safest for you to avoid using idioms in formal writing.

Hola Jacinto: As far as I know... Idioms y quotes son cosas totalmente diferentes por definición, aunque pueden cumplir misiones similares en la comunicación. Como seguramente sabes, unas son expresiones idiomáticas (dos o más palabras que unidas cobran un sentido particular o metafórico, diferente al que tienen por separado, y que en muchos casos no pueden traducirse literalmente a otro idioma. Modismos). Las quotes son simplemente citas: frases celebres, generalmente atribuibles a alguien también célebre. En ambos casos, igual que ocurre en castellano, su uso solo está limitado por el sentido común. Nunca usarías un modismo que pueda resultar vulgar en un contexto muy formal, ni emplearías una cita rebuscada en una conversación de bar.

I don't really know, what Jacinto said, but idioms and quotes are completely different things. A quote is something somebody else already said. Like "Don't cry because it's over, smile because it happend" is a quote by Dr. Seuss. And idiom is a pecularity of a language. Like when they say in English "it's raining dogs and cats". And of course it doesn't and directly translated into another language it would just be rubbish.