According to Carnegie, what were the benefits of capitalism over socialism?

Carnegie saw the benefits of capitalism through the lens of how it can benefit select individuals. Carnegie was not interested in exploring how capitalism exists for a large mass of people who are exploited in order to benefit the owning class. Rather, Carnegie saw how capitalism created freedom for wealthy people or for people who were willing to exploit others to become wealthy.
This is the aspect of capitalism that many find so alluring: one can, in theory, be born poor and become rich. The concept is that under capitalism anyone can reach their fullest potentials and dreams; however, it is a hard truth that this dream is often achieved through exploitation. In order for someone to become wealthy and obtain surplus, they must force scarcity and poverty upon someone else. Carnegie would never have become part of the super-wealthy class if he hadn't built his empire off of paying people measly wages and stealing land from Indigenous peoples.

Andrew Carnegie, the great American industrialist, had derived enormous personal benefit from capitalism. So it's not surprising that he was such a passionate advocate of the system that had made him one of the world's richest men. In his vigorous defense of capitalism, Carnegie looked at the bigger picture. He accepted that not everyone will succeed in the capitalist market place and that certain individuals will suffer. But ultimately, Carnegie thought this is all for the best. In keeping with his Social Darwinist views, Carnegie believed that the species as a whole benefits from capitalism, even if a few of its weaker elements end up going to the wall:

While the law [of competition] may be sometimes hard for the individual, it is best for the race, because it insures the survival of the fittest in every department.

Harsh though the capitalist system may appear to some, Carnegie argued that it provides the very foundations of civilization itself. He was scathing of his socialist opponents, accusing them of wanting to destroy those foundations. Only the strong—which of course would include Carnegie—are capable of building any civilization worthy of the name. If the socialists take the strong out of the equation and try to build a new civilization from scratch, they will surely fail. From his point of view, not only was this because socialism is a less efficient economic system, but more importantly because it seeks to turn the natural evolutionary order upside-down, putting the weaker members of society ahead of the strongest.

Carnegie conceives of capitalism as the engine of the world's industrial and social progress. He prefers capitalism over other economic regimes, including socialism, because he believes it generates a hierarchy of wealth and ownership where other regimes would only perpetuate poverty. Carnegie believes that when most of the new wealth is pushed to the upper class, the individuals at the top will know best, by virtue of their previous economic successes, how to allocate the excess wealth to social programs that lift up the poor.

Carnegie also believes that capitalism creates a competitive market, while socialism squelches all competition. When different firms and individuals have to compete, they naturally innovate and invent. Socialism, in contrast, provides little incentive for innovation. Though it can be argued that Carnegie's understandings of capitalism and socialism are incomplete, he tried to embody his capitalist ideal, using his steel fortune to invest in education and enrich public welfare.