Here's another video on it that goes more in depth: https://youtu.be/CJXp3UYj50Q
Basically the SteamOS version gets better fps in most games while also have way better battery life. Lower power games would get 2x the battery life when fps when both versions were targeting the same settings/fps.
Basically windows naturally has more overhead than Linux, which really impacts performance/battery life in low power games. But vulkan also typically outperforms directX, so many directX see a noticable performance increase on SteamOS as well (although some amount of this performance gain can be had on windows as well by using dxvk).