VCs will allow you to verifiably specify your date of birth or maybe your passport number.
What ZK does is allows a third party to ask questions like "is the date of birth of this person prior to 2-Sep-2006" (ie, are they over 18) or "is this person a passport holder for country X" and the ZKP system can say yes or no without disclosing the actual birthdate or the passport number.
It's is a real improvement in privacy, although I'm unconvinced it is worth the incredible inconvenience of implementing it.
