Humans are animals. There is no credible distinction such as that which you suggest.
Computers, as they work today, are not going to be aware of what they are thinking because they don't think. They follow instructions. A computer is a vast series of switches which are "flipped" on the basis of the settings of other switches. Think of them as being very much 2-dimentional. Actual thought appears to require something mimicking a neural network where some components are capable of monitoring and analyzing the states of other components within the network as the processes of the network progress. Think of a neural network as being more 3-dimensional.
Human brains are animal brains and function in precisely the same manner. The difference between a human brain and the brain of a chimpanzee is primarily the number of repetitions occurring during the development of the brain. As an organism developes, there are genes which determine the number of times the first neural cell will replicate, and the number of times each resulting neural cell will replicate. Thusly, this follows a standard binary sequence of growth, (i.e. 1 - 2 - 4 - 8 - 16 - 32 - 64 - 128 - 256 - 512 - 1024 - 2048...). The difference between the chimp brain and human brain is indicative of two extra replication sequences in the human.
All animal brains are made up of neurons. And at any reasonable level of analysis, a neuron is a neuron is a neuron. You can't isolate the way a human neuron functions from that of a chimpanzee or a sea slug, for that matter. The difference is in the number of neurons, not the functioning of those neurons. The key is in the genes passed to each species from its parent(s). Take a single neuron and set a gene to cause that neuron to undergo a dozen rounds of cell division and you have enough neurons to run a sea slug. Change that single gene so that the neurons undergo 25-rounds of cell division and you have enough neurons to make a human brain. Stop 2-3 rounds short of that and you have a chimp brain.
Sea Slug - Neurons 2^12
Chimpanzee - Neurons 2^23
Human - Neurons 2^25
(NOTE: While the reference to the genetic function which determine the number of neurons is accurate and the article from which the information came does claim 2^25 as an accurate representation, most sources indicate human brains contain closer to 100 billion neurons, (approx. 2^37), which seems more accurate. The point here to focus upon then, is that all one must do is alter the gene which determines the degree of replication for the first neural cell to create the difference between a human brain and that of another animal.)
Your division of human and animal brains simply doesn't exist. We are animals.
(Source: Discover Magazine, April 2006)