I have no idea how the various pathways of sensation, movement, emotion, etc are programmed in animals other than humans, so this is purely an attempt to explain some of the differences in sophistication.
Genetically speaking, we're not that different from a chimp/gorilla, and as it turns out, humans only have ~30K genes anyway, which, in the grand scheme of the animal kingdom, is not much, and it's certainly many thousands less than what scientists expected to find when the human genome project began.
So a lot of the difference between a human brain and, for instance, a chimp's brain, is actually purely numerical: the average human brain just has way more neurons than the average hominid's.
But is the fact that human neurons undergo a few extra cycles of mitosis before birth so significant? Yes! Nature plays a risky game–it only provides the foundation for brain function, and relies entirely on environment to provide the rest. Neurons are constantly creating and retracting synapses during any sort of learning process, and memory/retention actually depends on structural changes in neurons themselves (it is literally possible to distinguish a piano player from a violin player based solely on the superficial anatomical appearance of their brains). The advantage that humans have with so many additional neurons is an increase in the number of possible connections–several orders of magnitude more, in fact.
If the issue of quantity vs. quality strikes you as a counter-argument, consider this story (inspired entirely by Professor Robert Sapolsky of Stanford's bio dept): when world chess champion Garry Kasparov first played Deep Blue, he positively wiped the floor with the computer's circuits. IBM refined their darling product by basically teaching it every possible move, and when Kasparov played Deeper Blue, he was soundly defeated. Afterwards, he was apparently asked to comment on how a mere computer, just by knowing more moves, could possibly win in a complex scenario like chess. The story goes that he replied, "In quantity, there is quality". Numbers do matter! 🙂
Hope this is along the lines of what you were looking for–if anything remains unclear, please sound off! 😀