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.
In Coda we use zero knowledge proofs to stand in for downloading / checking the blockchain. This means you get the identical computation to normal blockchain syncing, but in only the size of the zero knowledge proof and your account, which ends up being both constant and ~20kb. Which makes a big difference if you want to use Coda from a phone or browser.
Biggest difference is Coda's constant 20kb vs still at least a few hundred megabytes for Vault. This matters because it helps in making cryptocurrency and cryptocurrency apps usable by people from their phones and browsers, at 20kb with Coda that's actually possible.
I haven't looked particularly into if any consensus / security assumptions are different either with Vault, but that could be another place for them to get down to a few hundred MBs without zero knowledge proofs.
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.