DirectX is more than just Vulkan. It does sound, input, etc...
Vulkan is like Direct3D 12, a low level 3D API. Between the two, most seem to consider Vulkan the better option. However, Vulkan has the reputation of being verbose and very much not noob friendly. It is mostly geared towards advanced engine developers who want full control to make the most of the hardware.
Besides 3D, the rest of the multimedia API are a bit of a mess it seems. On Windows and elsewhere. I haven't look at it for many years though.
For the two middlewares Unity and Unreal, on real applications, DirectX 11 will have better latency (lower CPU time mostly), DirectX 12 performance will be higher throughput (greater FPS), but neither will be by very much. Like a single application on ordinary hardware, it won’t matter. But for the thing I measure, occupancy, you can get something like 3x as much efficiency with DirectX on Windows compared to the same application on Vulkan on Linux.
