Electricity burned is the same. But so is the electricity burned while gaming, your system is limited by your GPU in both instances. If you have an RTX 5090 then you can throw almost 600 watts continuous at both which could use like 14kW/day, but most people will have more modest hardware, and therefore the energy usage of running a local model wouldn’t be all that much either - I don’t think many people limit their gaming because of how much the electricity would cost.

But when you have a datacenter with hundreds of servers capable of maxing out the residential house mains of half a dozen houses each (single NVIDIA NVL72 AI server has 72 GPUs and uses 120-150kW at full load) they can throw way, way more compute at you than you ever could manage locally.