Unsurprisingly, as an atheist, I rather enjoy my perspective on theism and the cognitive benefits this brings not only to my belief system in general but more importantly to the entire mental framework from which I view existence and reality. I therefore believe that there are psychological benefits to being an atheist and that these are not afforded to the theist. Obviously there is a certain bias behind this, as I am an atheist, and for me there are certain positive attributes that come with being an atheist - hence why I am one but this is a causal phenomenon common in the adoption of any mental schema. However I am only an atheist now. I once, rather embarrassingly, was a theistic believer. This means that I at least have a previous psychological state that I can compare my current one to and the cognitive grass is definitely greener on this side.
I shall discuss my world views from the theistic past to the atheistic present and demonstrate the cognitive benefits of atheism in comparison to theism. The first is the most profound. This is the fact that being an atheist produces less cognitive dissonance than being a theist of any type. The world, existence and general human and physical understanding is more rationally comprehendible as an atheist. However the theist fails dramatically when it comes to cognitive dissonance. Indeed theologians have spent most of history making self-justifications, arguing away contradictions and dealing with precepts that are by definition illogical. When it comes to mental acrobatics, the theist is Olympic gold medalist. One only has to look at the bewildering, painful and sometimes amusing manner in which theists attempt to argue that a benevolent God can allow or enact evil when in the pantheon of logical definitive accuracy the game is already up due to the immediate contradiction encountered when conjoining the two precepts.
What can the atheist say so easily that the theist must struggle so conspicuously to counteract? Well it is rather simple; the world simply looks like it does if there was no loving God. If you deny the existence of an omnipotent, omnipresent, benevolent God, and regardless of your position on naturalism, you would expect the world to look like it does. The universe, or universes, general existence, nature, its indifference and its chasm of mental agency allows you to reconcile the apparent fact that you cannot blame nature for outcomes it has no conscious choice over. Natural existence is indifferent to what is in the grand schemes of things petty social affairs carried out by one form of advanced animal species on one planet. Natural existence is not conscious and therefore not consciously responsible for the mind blowing cruelty and suffering of any biological organisms found anywhere in the universe.
In the face of this, the theist has an illogical contradiction to surmount. Not only does he argue the existence of a creator - itself deeply problematic in metaphysical terms but a loving one too. The latter is even harder to justify than the first argument from creation. Justifying evil in a form of existence wholly controlled by a loving God tends to be cashed out in terms of the argument that we cannot have love without evil and suffering; that we cannot know good without evil but if God is perfect he has failed to evoke a master plan that allows us to have good innately in itself without evil and in this respect God can keep his supposed master plan which is apparently perfect but yet untenable to mere mortals.
As you can see the atheist here is free of all tension and cognitive dissonance. Malevolence and illness in the world is more simply and rationally explained in light of atheism. As a consequence we also don't have to doubt God's seemingly bizarre and erroneous intentions such as why must we suffer? Does God punish through suffering? Why not skip mortal existence and go straight to the eternal afterlife? Such mental unrest leads to another point where the atheist's position is far more therapeutic than the theistic position and this is in terms of liberty, responsibility and control. This is because we are free from a sadomasochistic God who we should fear but also love. We can also live our life as we want without fear of being punished for misdemeanors or acts of hedonism. Where terrible crimes are conducted, humanity as a collective is forced to look at itself, understand the potential malevolence of human kind and seek remedy, prevention and progress where possible.
With rationality, self-ownership and liberty to purse multiple avenues humans are also free to discover more about themselves and the world around them; just ask Galileo but unfortunately you can't. Both the natural and social sciences have made considerable strides because we have endeavored to go into areas outside theism. Rationality and logic in philosophy has produced great arguments because there is no necessary religious restriction and in the world of natural science the scientific method has produced medicine which is far more useful then proverbial and erratic superstition.
When it comes to the atheistic worldview, logic, rationality, empirical demonstration or just simple philosophical argumentation, being an atheist provides very little cognitive dissonance in comparison to theism. Moreover the effects of this general cognitive operation not only means that we can look at social and natural phenomena in new ways that previously looked baffling under theism but also explore avenues that bring us new knowledge and rational explanation not found in theism. For these reasons, and perhaps many others that atheists themselves have found, the psychological benefits of atheism are rather staggering in comparison to theism.