Norton 360 doesn't have a free version, but ultimately it proves to be a more robust service.
The most significant selling point of the Avast Antivirus is that it has a free version, which provides basic protection. They just expand on the LifeLock functionalities. Higher Norton branches don't really add to the whole 360 suite. There are also even higher tiers with added LifeLock identity theft protection, a completely different product. Norton 360 doesn't have a free version, and its paid editions are Standard and Deluxe. There's also Ultimate, but it adds separate products: Avast Cleanup Premium and Avast SecureLine VPN. Avast Antivirus splits into Free, Premium (single-device), Premium (multi-device). How many features a service has heavily depends on which edition you're looking at.