It's still not clear, yet.
AMD Mantle will be multi hardware (AMD, Intel, mobile) and multi platform (win 7, 8, 9, linux), but the support behind is not as big as M$.
DX12 will be Win 9(10) only, multi hardware (AMD, Intel, nVidia, Win mobile) and the support will be continuous, it's M$.
It's known the AMD provided part of Mantle code for DX12.
Performance gains on Mantle are great, but not so consistent across hardware configurations. DX12 who knows, there's only one report as far as i know.