It’s more complicated than that. Zigbee switches and Z wave devices connected directly to the hub are using local communications, but the logic for many automations runs in the cloud. At the present time some automations using the official smart lighting feature and some automations using the official smartthings home monitor feature are eligible to run locally if all of the devices involved are using stock device type handlers which are also eligible to run locally. The only Wi-Fi integration at the present time which is eligible to run locally is with the Phillips hue bridge.
Even so, there’s a lot of stuff which doesn’t run locally. You can’t arm/disarm STHM locally. You can’t change the mode unless the cloud is available. You can’t use the smart things mobile app to control devices unless the cloud is available even if the phone is on the same Wi-Fi as the hub. (They didn’t have to design it that way, but they did.)
No custom code can run locally, so if you are using a custom device type handler, a webcore piston, or a custom smart app, that won’t run locally regardless of what the devices are.
So…You could have a zigbee device using a stock device type handler eligible to run locally on a time based schedule created in the smart lighting feature and that might be able to run without the Internet. But you wouldn’t be able to turn it on and off with the mobile app.
So, yes, Zigbee device might allow you to do more than a Wi-Fi device if the Internet was not available – – but only some actions and not if you are using a custom DTH.