The difference is that if you shut down gold mining, existing gold would still retain all its value. If you were unable to protect your gold reserves, they still retain all their value. If you shut down Bitcoin mining it's immediately worth nothing. In fact if you reduce your energy expenditure below some unspecified threshold it becomes utterly worthless. Mining gold consumes resources, but once extracted it does not. I would consider it to be zero-sum, as compared to the negative-sum nature of Bitcoin.
I would argue if institutional demand and even jewelry demand for gold fell, it would still be a fairly valuable commodity, and its use in electronics would likely expand substantially as it became less expensive. Connectors, switches, wires, PCB contact plating. All big demand drivers, and gold is better than the status quo - just too expensive at the moment.