I don't see that nein there could be really substantial. As for me, it just reinforces the negation begun by nicht nur before the comma:
Du grünst nicht nur zur Sommerzeit,
Nein auch im Winter, wenn es schneit.
I'd say the nein is there in order to keep both lines eight-syllabic. If you put sondern there, instead of nein, you get a syllable too much. And since in poetry you have lot of freedom, you can join both lines with an interjection-like nein. I cannot find more instances where nein could be relpaced by sondern.
The translation of aber coincides in some languages with the translation of sondern. But they're different words, even in those languages.
Examples. French mais is both sondern and aber. The same happens with the Russian words но and a; both are sometimes aber, sometimes sondern. So is but, in English. On the other hand Spanish has different words for that: aber is pero and sondern is sino. (And if you mix them, as many Spanish learners do, natives of languages where aber and sondern are translated to the same word, you'd be wrong.)
But in German it would sound strange to say (at least, outside poetry)
nicht nur zur Sommerzeit, aber auch im Winter [...]
After a nicht nur one expects sonder auch.
The rest of the answer is about English, actually, because one cannot explain here that but is not a word in English, but lots of words. Depending of it's role, you get a translation in German. But the key is that but in the meaning of although is aber; on the other hand, but, meaning rather, is sondern. A detailed answer would require rather expertise in English (not my case), not in German.
But the next examples should clarify.
Wrong use of aber instead of sondern:
- Nicht meine Mutter, aber mein Vater braucht Hilfe. (incorrect)
- Nicht meine Mutter, sondern mein Vater braucht Hilfe. (correct)
Wrong use of sondern instead of aber:
- Ok, wir fahren dahin mit dem Wagen. Sondern du musst fahren. (incorrect)
- Ok, wir fahren dahin mit dem Wagen. Aber du musst fahren. (correct)