the sky today is grey
What the difference between "dziś" and "dzisiaj"?
No real difference, just a stylistic one.
Well, stylistic difference is real and important too! Is "dziś" more spoken variant and "dzisiaj" more literature, or vice versa? Or it's not that point at all?
With those two, both seem completely normal, everyday language. So 'stylistically' maybe one syllable less or more could sound better in a given sentence. Most people wouldn't care.