Whereas the Proof of Replication is run once to prove that a miner stored a physically unique copy of the data at the time the sector was sealed, the Proof of Spacetime (PoSt) is run repeatedly to prove that they are continuing to dedicate storage space to that same data over time.
The Proof of Spacetime checks just a subpart of the data, running a probablistic proof on random locations to show that the miner has the specific bytes that should be there. Rather than checking for the original data, it checks the encoded version (the CommR), which during the Proof of Replication was proved to be an encoding of the original data. Since the CommR is a type of merkle tree called a Directed Acyclic Graph (DAG), this can be done with a merkle inclusion proof.
When miners agree to store data for a client, they're required to put down collateral. If they fail a Proof of Spacetime at any point during the contract, they'll be penalized. This is a key part of the incentivization that encourages good behavior amongst all the players in the Filecoin network.