Rand differs from Rawls' Theory of Justice in a very fundamental way on this. In fact I would argue so different as to be opposed. Rawls is essentially presenting a justification for wealth redistribution for the purpose of achieving "greater equality" and using an ethical argument to get there, while Rand presented arguments opposed to welfare statism. Her viewpoint was that ensuring individual rights were protect was the only way to achieve equality. She wrote in "Virtue of Selfishness" that "In a fully free society, taxation—or, to be exact, payment for governmental services—would be voluntary." Doesn't sound much like Rawls does it?
The stark divergence comes in their opposing fundamental principles. Either you believe in private property (based on self-ownership) or you don't. Either you believe theft is wrong or you don't. Either you believe the government is exempt from ethical wrongdoing (like theft) or you don't. Rawls' viewpoint was that sufficient human suffering (via poverty, income inequality or some vague natural level of social injustice) justifies theft, and Rand had the opposite viewpoint... that no amount of human suffering could make any theft justifiable. She wrote in Sept. 1969 of "The Objectivist", "suffering is not a claim check, and its relief is not the goal of existence."
Why is Rawls taken more seriously?
This is a harder question to answer, but I would offer a few reasons that I think are certainly true... there are other great reasons as well. I think it boils down to understanding cultural ebb and flow of the time and how power centers gravitate toward views that confirm their behavior. The first being that political elites like being told they can do unethical things (in this case theft, in some cases, drone strikes on American Citizens or using the IRS to target "enemies of the state", or conducting mass surveillance for "national security"). The second is that when they can market their unethical actions as "promoting equality" it feeds their narcissism and enables them to feel good about their unethical actions. Equality isn't something a lot of people argue against.
It makes political elites feel good about themselves.
Since Rand's viewpoint is entirely opposed to the ruling political elite who pride themselves on how much they can expand their budgets every year, what committees, programs, and special interests they can fund... which lobbyists they can pay off. They don't like being told that they are morally wrong for stealing from their citizens (politicians tend to be a tender, vengeful, bunch... and generally awful disingenuous people). So you have an environment across the country (federal, state, and private interests) where vast numbers of people are trying to co-opt government power centers to force agendas through the political system (they don't want to limit the power, they want to harness it), and you have megalomaniacs in power who love wielding it. When someone is saying "don't steal" is it any wonder they are completely ignored?
In order to redistribute wealth they know they first need to confiscate it (steal it), and they are more than thrilled that Rawls gave them yet another moral justification to do so. Things that political power brokers tend to like seem to be encouraged throughout history, regardless of how bad they are for the citizenry. An example would be Just War Theory, which basically boils down to "if we say so for any reason whatsoever." I'm reminded of Wesley Clark, in his now famous confession demonstrating exactly how Just War Theory works in practice after 9/11. Other examples that could be argued which fall into the category of "pro-government blank-check theories" are things like "independent" (yet appointed) Central Bankers which support "the regime" that appointed them via monetary policy. The entire body of Keynesian Economic theory (and what it has evolved into) of deficit spending and stimulus (which of course is working out great for us now...). Turns out politicians like being told they can just print infinite money, who would have thought?
Historically and culturally, a suffocating egalitarian shrink wrapping has been encircling the United States throughout the 20th century, and found anxious supporters in media and on college campuses hungry to champion causes and correct injustices (It's no different today). As political Marxism died out in the U.S.S.R., Cultural Marxism flourished in the United States through the sexual revolution, feminism, the founding of the NAACP, and the passing of Civil Rights (all of these things Rand was opposed to, but that is beside the point).
The culture of "promoting equality" in the United States steadily shifted away from a 19th century emphasis on equality through asserting and defending individual rights / individual liberty, to the 20th century of ensuring equality of outcomes (New Deal, Great Society, Federal Reserve mission of price stability) regardless of the circumstances that contribute to those outcomes (some being deserved, others not). It's known that Marx was influential on Rawls because he lectured extensively on Marx, but not on his economics. Rawls lectured on Marx's philosophical anthropology. His lectures generally took a full two weeks out of twelve, on Marx. By the time Rawls published in 1971 an explosion of interest in Marx's writings was underway, and Rawls was popularized because he is an ideological descendant (heavily influenced by) Marx. He wrote on achieving greater social justice, which folded in nicely with the cultural and academic trends preceding the 1970s. So, sad truth is Rawls was paid more attention because it was trendy, and most viewed notions of equality being achieved by individualism as "outdated."
Also, last quick note. Here is a good podcast on John Rawl's by Columbia PhD Historian Tom Woods.