No, it will not. Version of the Direct3D API alone has about as much effect on the game's appearance as version of Windows (= none).
Mantle is a low-level API that currently works only with certain HD7xxx+ AMD cards. Because it's specific to AMD, I don't see many developers adopting it. Because it's low-level, it makes less sense to deploy in on PCs. Different PCs have different HW and different HW would probably require different rendering code to achieve good performance. At last but not least, it's currently available only on Windows.