"She is not my sister but rather my mother."
Translation:Ella no es mi hermana sino mi madre.
I believe that the difference in usage depends on whether the second clause is negating the first. You would use "pero" when the second clause does not negate the first but rather adds on to the first clause. For example... "No soy española, pero hablo bien el idioma."
You use "sino" when the first clause is negative and the second clause is negating it. For example... "Hoy no voy a estudiar español, sino matemáticas." The meaning is more "but rather" than just "but".
However, I am not a native speaker so please correct me if I am mistaken.
I tried to use pero and sino and was corrected to "Ella no es mi hermana pero si mi madre." *Si with the accent, as in "yes." This was confusing to me, but I had it explained and it appears that Duo is saying that I CAN use pero if I want, but since the phrase uses "but rather" I have to have another word to compliment pero. Clear as mud?
As AliciaGravez says above, pero means but, and sino means but (rather) or but (instead). Sino is used in the second part of a sentence when the first part is a negative clause, and the second part is a phrase that corrects it.
No bebo té, sino café. I'm not drinking tea, but (rather) coffee.
Nuestra casa no es verde, sino azul. Our house isn't green, but blue (instead).
This seems to me to be a pretty good explanation: https://www.realfastspanish.com/vocabulary/pero-vs-sino