The obvious thing is performance. Vulkan runs better which means more fps but it really is a small amount. Vulkan is supposed to be more finished and constantly updated vs DX11 which is now supposed to be replaced by DX12 (though it seems unstable in most games).
AMD states that "Vulkan supports close-to-metal control, enabling faster performance and better image quality across Windows 7, Windows 8.1, Windows 10, and Linux. No other graphics API offers the same powerful combination of OS compatibility, rendering features, and hardware efficiency."