It appears that what really differentiates the 2 hemispheres is the type of attention that they bring to the world. For very good evolutionary reasons, the left brain brings a very detailed, sequential and abstracted view to things that we already now to be important, like food, and ignores the unfamiliar. The right brain, in stark contrast, brings a much broader, uncommitted, holistic view and focusses on the new, the different and the incongruous.
These 2 types of attention are both vital, and are so important that the corpus callosum (the bit that bridges the 2 halves) functions to inhibit one or the other hemisphere allowing the currently active hemisphere to get on with its job whilst the other is pushed into the background.
Birds and other mammals have these left / right distinctions too, so clearly they are of enormous evolutionary importance. A basic example is this, whilst the left brain is bringing its very detailed focus to bear, perhaps on food we have found, in the background the right brain is broadly vigilant for threats, preventing something making a meal of us. So effective at this is the right brain that we can have often commenced evasive action before we are consciously aware of the threat.