"No son semanas, sino meses."
Pero is used to join two contrasting ideas when the second phrase does not negate the first. Instead, you can think of it as adding on to the first idea.
Sino, on the other hand, is used generally in negative sentences in which the second phrase negates or corrects the first. The equivalent in English would be "but rather" or "but on the contrary".
I have to say i've consult the 'rae' for this, and the explanarion is (hope i tanstlate correctly) pero: to express a concept different or give more information on the thing exposed sino: to express somethjing negative againt something positive If it is not clear let me know.
Sino is an elegant way to provide a non-obvious alternative to a negative statement, without having to start a new sentence.
To avoid confusing it with pero, try replacing it with although–if it works, it's pero; otherwise, it's sino.
> — Me apetecía verte hoy, pero hasta mañana no puedo.
> "I felt like seeing you today, (but/although) until tomorrow, I can't."
> — Pues yo no contaba con verte hoy, sino mañana, así que perfecto.
> "(Well) I wasn't counting on seeing you today, (but/I was counting on seeing you) tomorrow, so that's perfect."
I go deep into the usage context of sino in this NachoTime post.