CODA uses cryptographic arguments to demonstrate that a block is valid (including ancestor blocks). I haven't read much of the Vault paper, but it sounds like they're having users vote (with weights proportional to their stake) on whether blocks are valid.

If someone corrupted enough voters, they could theoretically create a "Vault certificate" for an invalid block (e.g. one which spends the same coins several times), and other nodes would accept it. That wouldn't be possible with CODA, unless someone broke their cryptographic primitives.