This is a very huge question spanning multiple fields in philosophy. I do not have the expertise to cover all of these, so I'll focus on my personal favourite, the Philosophy of Mind aspect.
As it stands there are no universally agreed upon answers to whether humans are different from computers in how they think. There are people on both sides of the argument, and there are multiple positions on either side.
Consciousness Exists, and We are Different: Cartesian Dualism
One of the oldest positions that strongly implies a difference between humans and computers is that proposed by Rene Descartes in the 17th century. This is roughly the standard traditional substance dualist view, which says that as humans we have both a body and a soul, and that the two are decidedly different. Specifically, while the body is made of matter (in the intuitive sense), the soul is completely intangible and not expressible in physical terms. It exists outside of our physical world but can control our body by providing input to the brain. It also in the soul that our conscious mind resides, and thus consciousness becomes a non-physical phenomenon. Thus in Cartesian Dualism, even if you make a machine that functionally replicates the human body, it will not be conscious. Cartesian dualists (who are becoming rather less common these days) therefore hold that there is a fundamental difference between machines and humans, which stems from our having a soul.
There are several criticisms of Cartesian Dualism, and the most threatening is the Mind-Body problem. This is the question of how it can be that an immaterial soul could possibly have any effect on the physical world: how does our mind drive our body, if the latter is physical and the former is not? No certain mechanism has yet to be proposed (Descartes believed it was the pineal gland in the brain), and by the inherent nature of substance dualism it seems very difficult for any theories to be rationally necessary. For the most part belief in Cartesian Dualism boils down to faith; after all, how on earth could we go about scientifically validating the means by which something by definition unknowable to us changes the physical world? This is generally why Cartesian Dualism is a bit of a weakening position.
Consciousness Exists, but We are the Same: Some Non-Reductive Functionalism
There are a huge number of functionalist that propose no inherent difference between humans and machines, even though we have consciousness. For brevity I will present probably the most common such position, which has been developed significantly by David Chalmers. Basically the theory is that consciousness is a non-reductive consequence of certain functional structures. Simply, if you build a machine that works in the right way (and the human body is one such machine), it will have consciousness. Chalmers does not think that consciousness can be reduced to individual physical pieces, but rather that consciousness itself is a sort of "fundamental particle" that is created from the right sort of structure.
Of course there is the difficulty of figuring out just which functions lead to consciousness, and how they do this (this has been called the "Hard Problem of Consciousness" and "the Explanatory Gap"). Some believe that this is a resolvable issue while others think that, similar to the mind-body problem, it's perhaps impossible to find a definitive answer.
Consciousness Does not Exist/is Nothing Special, so We are the Same: Some Reductive Functionalism:
This is a position primarily developed by Daniel Dennett and Susan Blackmore. The argument is effectively that consciousness is nothing more than an illusion caused by what is called "access consciousness." Access consciousness is the aspects of our mind that are almost definitely understandable in reductive, functionalist terms. This includes our ability to sense and react to environmental stimuli, report our personal mental states, and other similar abilities. Most agree that we will eventually be able to explain these through traditional reductive cognitive science and neuroscience, and people such as Dennett and Blackmore believe that our own consciousness is nothing more than an illusion resulting from these abilities: there is nothing particularly special about it. Thus there is no reason machines would be inherently different from us; build them the right way, and as a result of the reductive laws governing their access consciousness, they will have consciousness in the same way we do.
Unfortunately this is an incredibly counterintuitive argument and just as unverifiable as Chalmers' position: even if you did have a theory of exactly how access consciousness worked, how could you possibly verify the existence of the consequential conscious experience when it's purely subjective?
Conclusion: We have no Idea
What I've said above does not even come close to coming close to scratching the surface of all that has been said - it summarizes a fraction of an introductory philosophy of mind class which itself barely exposes the vast literature out there. However what I hope I've made clear is that there are no logically rigorous arguments for any particular position out there. They all fundamentally hinge on some sort of intuition or faith (e.g. Chalmers' Zombie argument depends on his intuition, Dennett's counter argument on his own). Anyone who says that they know for certain the answer is either lying, fooling themselves, or hiding some brilliant insight.
Whether with regard to our mind we are the same as computers or not thus is an open question with many developing answers. You should read all of them, and if you're confident, take your pick for one to work on :) Just remember that at best your position will be probable until some major developments have been made.
ReallyRational: "There is no definitive reason to believe that computers can't be conscious". There is: they are not organic systems, and consciousness has only been seen in organic systems. Certainly anything is "possible" and nothing is "definitive". But I am a scientist, not a philosopher: I base my conclusions on empirical evidence. Show me a conscious computer, or even a plan of how to build one.
Our empirical evidence tells us nothing more than that we ourselves are conscious. There is no evidence convincing me that any of you have consciousness. How would you even go about empirically verifying something which is by definition purely subjective? The only thing I seem to know for sure is that I myself am conscious, and even this has been subject to skepticism over the years. To be clear, this is not some esoteric opinion I have, but the generally agreed upon view, and certainly much more logically rigorous than trying to argue that other things have consciousness.
Really Rational: "but for all we know our impulses are determined by the functionality of our neurons, and it is only the complexity of our brain that veils this." We must judge based on what we know and see, not on what might be. "He who lies, speaks of things far away" - Talmud.