The Görli testnet is all about bringing cross compatibility across the different Ethereum clients.
Historically, the only cross client testnet was Ropsten, which used Ethereum's mainnet PoW algorithm. However, since the testnet holds no value and provides no incentive to miners, it was hard to maintain a strong and resistant network. In fact, the network was quite easy to attack, and malicious individuals had done so.
So, to solve this, parity-ethereum created Kovan which uses the Aura PoA consensus algorithm, while Geth created Rinkeby which uses the Clique coneseus algorithm. These two networks were not compatible with one another, and thus issues with one client would take down the respective testnet.
To solve this problem, the Görli project was started to implement a single proof-of-authority consensus algorithm across not only the two main clients, but all Ethereum clients. This brings the security of a PoA network with the reliability of multiple client implementations.
One reason to start a new network is to start fresh, and not have to deal with multiple years of testnet data which really isn't useful moving forward. In addition, the Görli network provides features not available on Rinkeby like the GoethBridge which provides easier access to testnet ETH.