Oxide is a game engine (like Frostbite, Unreal, or Iron); Mantle is an API.
Specifically, Mantle is an alternative to OpenGL or Direct3D. From what I hear, it is incredibly efficient, being able to use the resources of your graphics card more efficiently than
What this means is as follows: any game engine that is written to take advantage of the Mantle API can use Mantle to make very good use of (compatible) graphics cards.
For example, Battlefield 4 apparently runs 45% faster on the same hardware using the Mantle API than it does using Direct3D (http://bf4central.com/2014/01/battlefield-4-runs-45-faster-amds-mantle-software/).
I'm excited. Mantle API + new engine = win.