Capitalism is not about "free markets." In the economic sense of perfectly competitive markets, free markets do not and cannot exist in actual reality. We have a few markets that are approximately perfectly competitive, but those markets usually exist not to supply the consumer but to supply monopolistic or oligopolistic end-stage producers. For example, economists usually consider agricultural production to be close to perfectly competitive, but most agricultural production goes through one of the big food processing conglomerates, such as Archer Daniels Midland or General Foods. Similarly, a lot of clothing production is perfectly competitive, but most ends up being sold through Walmart, Target, or a couple of other chains.
In the political sense of markets free of government interference, free markets are impossible. Markets require both money and reliable promises, all of which require a government. Political free market advocates want enough government interference to protect their own interests, but not enough to protect yours.
If capitalism is not about "free markets," then communism as anti-capitalism cannot be about "central planning." Central planning is a tool that both communist and capitalist governments use under various historical circumstances, mostly when fighting wars. Most Western economies were very strictly centrally planned during the Second Imperialist War. Both the Soviet Union under Lenin and Stalin, as well as China under Mao, had to fight long, protracted, and continuous wars (especially Russia), and simultaneously had only a few decades to catch up with centuries of Western capitalist development (especially China).*
*The Soviet Union and China also developed communism in a social and cultural environment very different from that which we are used to in the West; because of this deep cultural difference, we must expect them to do things that we find absurd, incomprehensible, or repugnant, and would never be countenanced in the West. (I'm sure they find many of our practices absurd, incomprehensible, or repugnant.) Communism does not entail becoming Russian or Chinese.
I will readily admit that a centrally micromanaged economy is a Bad Idea in many circumstances. But not all, however: almost all capitalist businesses, some of them very large, operate internally as centrally micromanaged economies. Furthermore, at the macroeconomic level, investment is controlled by a fairly small group of people, who exogenously set various macroeconomic variables such as interest rate, money supply, bank reserve requirements, and bank lending standards. (The argument in even neoclassical economics is not whether to exogenously set macro variables, but which variables to set and which variables to allow to respond
The who is the crux of biscuit.
Capitalism is the idea that private individuals should "own" the means of production, and that individuals should use the means of production for their own benefit. In capitalism, social constraints on individuals' use of the means of production should be as little as possible; in practice, those social constraints are determined by the owners of capital themselves, to prevent self-destructive internecine strife. Ownership (and remember that ownership is socially constructed) of the means of production, i.e. money, grants individuals enormous social and political power. I would be very happy to let you control almost all the guns if I get just enough guns to control all the money. Mao was wrong: political power comes from the vault of the bank, not the barrel of the gun.
In contrast, Communism is the idea that "society" should "own" the means of production, and "society" should determine how to use the means of production for the benefit of workers. Of course, "society" is a placeholder here: individuals are real, "society" is an abstraction that should not be reified. It's a placeholder for a social process, including mores, customs, and institutions, that individuals socially construct. But communism is more than just some sort of social process; capitalism is also a social process which is itself socially constructed. Communism demands at least that the social process we use to determine the use of the means of production operate for the benefit of the workers, not the owners of capital. The argument among communists, then, is what kind of mores, customs, and institutions will actually benefit workers.
(I want to take a slight detour here. I advocate what political scientists would strictly term not communism but democratic socialism; strictly speaking, in political science jargon, "communism" means either the end stage of socialism, rule by the communist party, or a political system based on pure Kantian altruism. This labeling is applied by academic political scientists, who of course constitute a capitalist institution. I could call myself a socialist, but in reality, there are too many people who call themselves "socialists" that advocate (at best) only welfare-state capitalism, and laissez-faire capitalists like to brand as "socialist" anyone who advocates even the tiniest bit of welfare-state capitalism. ("I'm a socialist!" "Oh, you're voting for Obama?" Sigh.) I call myself a communist to make clear that, whatever my specific ideas, I advocate a radical departure from capitalism, not incremental welfare-state-ism; I'd rather have that distinction immediately clear and leave the details of my ideas to further discussion.)
Russia and China tried to see to the workers' benefit by implementing a self-selecting, self-promoting national communist party can exercise direct managerial control over factories, stores, farms, etc., and the workers who operate them. This effort collapsed in Russia, and it has been so substantially modified in China that, although controversial, one can legitimately call China an authoritarian capitalist state. (There's some evidence that the coordination afforded by an authoritarian state, which does not depend on economic coercion to maintain political power, operates more efficiently than a technically republican state such as the United States.) The two biggest justifications for this particular structure are that both Russia and China needed economic development immediately, and that they had few of the social, cultural, and institutional precursors for a successful republic or democracy. (A republic or a democracy, like any other social construction, cannot simply be dropped on a society like an overcoat.) If you're in a substantially economically underdeveloped (in relative terms) country with a long tradition of authoritarianism, a ruling communist party is not such a bad idea; the alternative is, perhaps, only a choice between tyranny and fascism.
But the West does not have the conditions that justify party rule. We have a highly developed economy; we don't have to make a "big push" to catch up with anyone else to secure our survival. We also have a the social, cultural, and institutional characteristics that can support the development of democracy. There's no reason we must make the same sort of compromises and adjustments required by Russia and China. (Neither should we ignore their accomplishments or, where appropriate, copy their methods, simply because they operated in a different cultural context.)
The point of communism, if we go to Marx, is that the workers should own the means of production. We communists are struggling, just as the capitalists struggled for centuries, to define what that means, and we have to abandon ideas that either (depending on who you ask) failed or are no longer applicable to material circumstances. In the West, therefore, communism has to mean democratic communism, ownership of the means of production by a democratic state.