First, evolution tends to innovate at the periphery. The new things that mark out a new species tend to be in the structure and function of its body (including the movements it can perform), its organs, and its sensory equipment. When it comes to the brain, the innovations are less specific, but tend to involve tweaking the general build plan used in this branch of the family tree: some parts of the brain grow bigger, some smaller, but the types of structure are the same. Evolution modifies the existing plan, more here, less there; it doesn’t build a new brain piece and add it in.
Take bats. They can navigate in the dark using sound. But there is no special new part of the bat-brain for navigating using sound. The ability to emit ‘ping’ noises, and enhanced hearing to differentiate the echoes that come back, are innovations. The bat brain uses similar types of brain structures to other mammals, but develops in them the ability to combine sound information to guide flight and not bump into cave walls.
What are the specific innovations in humans, separate from other social primates? We stand upright. We have hands and vocal articulators (lips, tongue, vocal cords) evolved to allow precise movements and speech production. Rib muscles that allow us to generate a smooth flow of air over the vocal cords to produce speech (this works fine so long as we’re not laughing). One particular part of the brain, the cortex, has grown bigger, giving us more thinking power. More on this later.
[1] Here are some fun papers on linking evolution and development to understand how the human brain works, by Barbara Finlay and colleagues: Endless Minds Most Beautiful. Developmental structure in brain evolution.