Privacy.

Every blockchain (bitcoin, bitcoin gold, bitcoin diamond), although anonymous, your accounts and transactions are traceable.  Just take a quick search on the blockchain explorer with your wallet ID and you can follow it, how much in the wallet, where it was some of the coins transferred, which wallet ID was transferred to, how much coins, etc..  If you attach a name, website, IP address, etc to your wallet (e.g. wallet on the coinbase.com), eventually people can trace it back how many coins are in the wallet and who it belongs to.

With zero knowledge proof, once your coin is in "private wallet" as opposed to "transparent wallet" or public wallet, then your coin is not traceable on the blockchain.  Once from send from a transparent wallet to a private wallet, wallet ID, amount, and others transaction-related information is placed in the 0 proof algorithm and completely hidden from blockchain and only exposed to 0 proof.  So its kind of like coins disappear to everyone on the blockchain else except whoever holds the private key can access.