Pieśń is more elevated. It concerns traditionally significant themes like fatherland, religion, fight, awe etc. Piosenka can be about anything including those, yet it's somehow easier and more approachable. Although there's no difference between them in English, we are rather strict about this distinction in common speech.

'Piosenka' could be a popular pop song or a ballad, etc. 'Pieśń' is rather serious song like maybe anthem or patriotic song, and also an old, religious, Polish songs are called 'pieśń' (pl. 'pieśni').