I suspect that what really humans apart from other animals is language. True, some other species do show a capacity for a crude form of it, but it's never anything like human language, or used to the same great extent. Language allows us to do a great many things that a mere incremental increase in brain size never could. It allows groups to be organized by verbal instruction, discussion, agreement, rather than just by instinct. It allows powerful sharing of information about immediate surroundings. Even more importantly than those two things, it provides a way for knowledge to accumulate over generations.
These things seem commonsensical, but there is a less obvious way that language seems to help us: it gives us an efficient way to deal with a world that is far to complex for our brains to fully comprehend. It lays the groundwork for mathematics, whose usage almost always involves converting reality into linguistic symbols before we apply its tools. Mathematics in turn allows for many scientific ideas, especially ones in modern physics, which would be impossible to comprehend without abstraction. On a more every-day level, it is very convenient to be able to refer to a complex system of millions of individual human agents with a short linguistic marker such as "New York," "Iraq," or "The European Union." Such simplifying markers can allow us to lose sight of the full nature of the things we are discussing, but it would be impossible to think on the scale we do without occasionally breezing past the details.
If the above is right, there is a very interesting philosophical question that comes up: what does it mean for a sentence that is in fact a gross simplification to be true? We use such sentences all the time and almost always think of them as just true or false, but that seems to ignore the imprefect nature of their representations. I'm not suggesting for a moment that truth is just a linguistic construct; of course I think that's absurd, but it would be worth trying to refine our ideas about truth to reflect the imprefect nature of language.