Often better performance is due to more efficient use of hardware, which comes at an energy cost. Other times it allows the hardware to work more efficiently, like avoiding spinning on locks.
The simpler explanation is that Windows is poorly optimized, and runs a ton of inefficient crap in the background that’s taking cycles away from the CPU and maybe even GPU.
It doesn't even have to be worse, just different
The Vulkan difference can actually be achieved on windows as well, by using DXVK on windows.
The overhead difference is most seen on the lighter games, like Dead Cells and Hades. It's why we see such massive increases in battery life for those games.
At that point it has to be a platform/stack difference.
On the flip side, the CPU may be doing less work due to things like more efficient drivers / less OS nonsense tying up CPU, which would also reduce power draw.
It's a matter of balance.
More efficiency means less power for the same computations. If you increase the computations more than your gain in performance, you use more power. If your efficiency gains are not fully utilised by increased computations, you use less power.
This would be in contrast to say wasting a lot of cycles in a spin lock due to thread contention, where reducing or removing the contention might gain you both performance and lower energy cost due to less waste.