This is a huge question, and is at the core of most Cold War history. There are gads of reasons for the Cold War, and I'd be happy to give you some of the arguments from big names in the historiography (Gaddis, Lafeber, Westad, Lundestad, Iriye, Kennan, et alia) but as you've asked basically why the two powers couldn't live and let live, I'll give you a few reasons.

Ideological: To shrug off the differences between what are, on the surface, simply different economic systems, is to ignore all the things which go along with those systems. Keep in mind the fact that Stalin gave a speech on February 9, 1946 openly saying that another war, this one with the capitalist nations (especially the U.S.) was inevitable. This is not mentioning entirely that Americans and their government had a basic fear and distrust of communism dating back to the birth of the Soviet Union. America never trusted the Soviets, and the Soviets only slightly trusted F.D.R., but never Truman or any leaders thereafter.

One should also keep in mind that in a post-Pearl Harbour, post-World War atmosphere, the most influential and numerous American leaders believed that America simply could not allow Europeans to direct world affairs any more - they were too irresponsible, short-sighted, petty, and immoral. (This was their reasoning, not my argument). Leaders and influential officials like Truman, Eisenhower, Marshall, Nixon, Dulles, MacArthur, Acheson, Lodge, Kennan, Nitze, and so on all vehemently agreed that America had to play the dominant role in shaping the new world order - each disagreed on what that order should look like but that's life.

On the Soviet side, you have a nation which has legitimate security concerns and a now wholly justified existential fear of a healthy Germany. It cannot and should not ever be ignored that on a political/diplomatic level, Russia has an almost innate insecurity about its borders. The problem arises in that Russia's solution to this has usually been to attempt to annex or dominate those regions by which it feels most threatened.

Diplomatic/Political/Politik: Think about the world just after the Second World War. Europe is effectively trashed. Germany, the nation with the most potential to be the solid economic hub to anchor Europe is divided into broken sections and occupied. France is attempting to claw back its fragile empire, and the British continue to see themselves as a dominant world power despite the most powerful nations of their commonwealth (India, Canada, Australia, New Zealand) moving away from them. In Asia you have a Chinese civil war raging after years of the nation basically being treated like an imperial stomping ground, Japan has been more or less flattened, and British, French, and Dutch imperial holdings are ever more unstable. While in no way a power vacuum, the world at this point (I'm ignoring Africa, the Middle East, and Latin America for the reasons that with the exception of Latin Americans, no one saw it as anything other than an American dominated region, the Middle East had been unstable [in the international sense] since the collapse of the Ottomans, and Africa wasn't really a contested target until later in the Cold War) was disordered and in need of stabilisation. The United States and the U.S.S.R. both saw themselves as the proper and sole entity capable of creating and maintaining that order. Both had legitimate security concerns - both had been attacked, the U.S.'s economy required a healthy global economy, the U.S.S.R. and Stalin understood the hate world powers like the U.S. and Britain had for them and while not cowed by the atomic bomb certainly felt that belligerent and well armed enemies surrounded them.

There is also the simplest most cynical explanation in that both nations were capable of expanding their power and influence, and thus they did so until they ran into each other. Having run in to each other their frustration turned into enmity which, since neither could feasibly knock out the other without suffering crippling losses (more so the U.S. than the U.S.S.R., which really couldn't have effectively fought the U.S. until at least the late forties) this translated into an angry stalemate, thus the Cold War.

In my opinion, the answer is a combination of these factors, with each and more feeding off of, influencing, shaping, and driving the others. Documents from the period clearly show that American leaders' sense of duty and responsibility made them believe that the world genuinely needed American leadership, and that communism and the Soviets were evil. And I mean genuinely evil. John Foster Dulles truly saw communism as the Devil's work. The Soviets genuinely believed that communism was the way to a peaceful future. Stalin thoroughly believed that capitalist nations would invariably go to war with each other, and without doubt would attack communist nations. Those Soviet leaders who survived Stalin's purges now had a series of paranoid layers in their psyches - being unable to trust your closest associates who are supposed to share the vision of a communist utopia with you means you are very unlikely to trust capitalists (especially those who openly state that your system is evil, oppressive, and counter to human nature).

I'm not suggesting that there was never any chance of accommodation, but when you combine economic and political interests, religious fervour, a sense of destiny, Real Politik, and a long standing distrust, the chances of friendly relationships are very slim indeed. And this is without really considering any other actions each side took in the post-war world.