What is public verifiability, and why is it the property that most sets a distributed ledger apart from a central system?

Public verifiability lets any observer confirm the system's state was updated correctly by the protocol — so everyone converges on the same view — instead of having to trust a central operator's word for it.

In a distributed ledger each state transition is confirmed by verifiers (e.g. miners), which may be a restricted set — but any observer can check that the transition followed the rules, and all observers eventually share the same view of the ledger (at least up to some length). In a centralized system, different observers can hold entirely different views and generally cannot independently verify that every state transition was legitimate; they must trust the central entity to report the correct state. Public verifiability is also what underpins integrity: if anyone can verify the state, anyone can detect unauthorized modification.