The words distinto and diferente mean "not the same". They are often confused, though having different meanings.
Diferente means "different", "not the same", "which varies", "not similar". It is used with objects of the same category, which are, nevertheless, different:
CORRECT: Mi hermana y yo somos muy parecidas, pero diferentes – My sister and I are very similar, but different
CORRECT: El piano y el violín son instrumentos muy diferentes – The piano and the violin are very different instruments
The word distinto means "(totally) different", "fundamentally distinct". It is used with uncomparable or basically different objects:
CORRECT: Mi compañera María y yo somos muy distintas – My partner Maria and I are very different
CORRECT: El piano y el armario son dos cosas muy distintas – The piano and the closet are two very different things
SOUNDS STRANGE: Mi hermana y yo somos muy parecidas, pero *distintas– you cannot say so because similar things cannot be totally uncomparable
Diferente can also mean "another one":
– ¿Conoces la película Solaris? – Do you know the film "Solaris"?
– Sí, allí sale George Clooney – Yes, with George Clooney
– No, te estoy hablando de una película diferente – No, I’m talking about another film
Diferente can also mean "various", and distinto, "special":
Me gusta conocer a gente diferente – I like to meet different people (various, not similar)
Me gusta conocer a gente distinta – I like to meet different people (not like others, special)
Both adjectives can be used with the prepositions de or a:
Eres distinto a los demás = Eres distinto de los demás