It is not a difference in degree, it is a difference in kind and seems to be unique to higher primates (you can learn a dog to do tricks, but it won't spend its time coming up with new tricks), one of the most interesting science videos I have ever seen is one where a bonobo learns to write a symbol that had been on a computer it used to express its emotions.
What spawns creative thinking and self-awareness is of course a mystery, but I think it's a consequence of creating a complex enough system and creating a system that's built on ordered chaos.
* http://en.wikipedia.org/wiki/Self-awareness#Self-awareness_i...
And it is a difference in degree. A dog frequently comes up with new, previously unseen behavior -- burying certain objects, shredding toilet paper, etc. The problem is that most behaviors are not seen as useful by humans, hence they go unrewarded as "tricks". But there are also examples of dogs learning useful behavior on their own, from learning to bark something that sounds like "I love you", to learning to knock down an owner who is about to have a seizure, to learning to fetch the leash when it wants a walk.
Defining "intelligence" as "doing something that humans can do that animals and computers can't" seems somewhat self-centered (although I admittedly cannot think of a better one off the top of my head).
From what I've gathered, the consensus, both in neuroscience and in philosophy of the mind, is that consciousness is totally emergent from the "simple" building blocks; there's no specific component unique to higher primates that explains the difference.