Despite both being advertised as privacy-focused cryptocurrencies, they are very different.

Summary

Monero uses ring signatures, RingCT, and stealth addresses to hide information on the blockchain. For every transaction, there is no way for an outside observer to determine the sending address, the amount sent, or the receiving address. Optional transparency can be provided off-chain by giving the desired person(s) the view key. Monero can be used with Tor or I2P manually, and Kovri is being worked on to provide native anonymous routing functionality.

Verge began under the original name DogecoinDark. It uses the same exact technology as Bitcoin and Dogecoin on the blockchain. For every single transaction, the sending address, amount, and receiving address are all visible. Verge users can optionally use Tor to send transactions.

It is disputed whether DogecoinDark began as a fork of Dogecoin or of NovaCoin/Peercoin.

Impact

Let's walk through a few use-case situations to compare how these coins protect the privacy and security of users.

Situation 1: Bob buys cryptocurrency on an exchange. Suppose this exchange complies with all local legal regulations and collects significant information about its users, including their address, IP address, name, and social security number.

With Monero, you can send coins off the exchange. The exchange knows what address you send the funds to, but it does not know how much money is in the address, and it does not know where the money will be spent in the future. Once it leaves their possession, the exchange doesn't know anything about it.

With Verge, the exchange knows what wallet you withdraw the money to, and it knows the balance of the account. It knows every other transaction involving this address in the past and in the future. It can track the funds sent throughout every address the funds touch forever.

Situation 2: Alice deposits cryptocurrency on an exchange. Suppose this exchange collects information as mentioned before.

With Monero, the exchange does not know what address the funds came from, and it does not know what this money was previously used for. It does not know the balance of this address.

With Verge, the exchange knows the address the money is coming from, the address's balance, and every transaction this address is associated with in the past and in the future. It can track the funds sent to the exchange throughout their past to learn exactly how they were transferred between different addresses.

Situation 3: Charlie buys coffee with cryptocurrency at a local shop.

With Monero, the shop owner does not know what address the funds came from, and it does not know what this money was previously used for. It does not know the balance of this address.

With Verge, the shop owner knows the balance of the address, and it knows how this money was previously used. The shop can audit the incoming funds to determine if they are "clean" or "tainted"; they can refuse payment of "tainted" funds or charge a higher rate. The shop owner can attempt to charge a wealthy customer more and threaten to steal their funds. It knows if this address is used in the future.

Situation 4: Dan receives payment from a stranger.

With Monero, Dan does not know the address of the stranger. The stranger knows Dan's address. Dan and the stranger do not know anything about their transaction history.

With Verge, Dan knows the address and balance of the stranger. The stranger knows the address and balance of Dan. They each know all transactions made with these accounts, including any possible transactions in the future.

A Note About IP Address Obfuscation

Tor and I2P can be used by pretty much any cryptocurrency, including Bitcoin and Monero. You do not need to use a separate coin to receive the benefits of routing traffic through Tor. These are separate things.

The IP address of a transaction is never stored on the blockchain. It is only leaked as metadata to nodes that you are connected to. For instance, suppose Alice runs a full node and sends a transaction. Suppose she is connected to 8 of 1000 nodes on the network. Only these 8 nodes receive her IP address, and they do not know for certain that they were the first to receive the transaction request. There is still a level of uncertainty.

Hiding an IP address is not nearly enough to have privacy. It must be combined with an obfuscated blockchain. If a user had to choose one over the other, they should prefer an obfuscated blockchain, since the blockchain information is shown to the entire network and stored forever.

A Note About the "Wraith Protocol"

Verge has been advertising the including of the "wraith protocol" in their coin for some time now. The "wraith protocol" is a rebrand of stealth addresses used in Monero and other CryptoNote coins. It has not yet been implemented.

Monero and other CryptoNote coins have had mandatory stealth addresses for their entire histories. Verge plans to make this feature optional, and NOT include other features such as ring signatures or confidential transactions.

This feature only really works if everyone uses it. It offers little to no protection if only a small fraction of people actually use it. The majority of the blockchain will still be completely transparent, which means there will be little or no privacy protection for those using the "wraith protocol."