This is, of course, largely a matter of personal opinion. There is no way to objectively say whether it is more correct to call these men robber barons or captains of industry. My own view is that they were both.
It is undoubtedly true that these successful businessmen like Andrew Carnegie and John D. Rockefeller acted in ways that warrant the appellation “robber barons.” This was a time when there was little government regulation of industry and there was intense competition to get ahead. These businessmen used any tactics they could to make their businesses more competitive. They competed ruthlessly and were quite willing to drive other companies out of business, often ruining the finances of the people who ran those other companies. They were also perfectly willing to exploit their workers, paying them relatively low wages and (as with Carnegie and the Homestead Strike) employing violent tactics to prevent them from doing things like unionizing for higher wages. They also did things like bribing government officials to do things that would be beneficial to their businesses. All of these things make them deserve the title of robber barons.
However, these men were also captains of industry. It is very hard to dispute the fact that their actions helped to build the American economy. Through their competition, they discovered new and more efficient ways to make products and to organize their companies. In doing so, they made it possible to manufacture goods for relatively low prices. This allowed ordinary Americans to buy more goods and enjoy higher standards of living than they previously had. In addition, men like Carnegie and Rockefeller gave huge amounts of money to charitable causes. For example, Carnegie put a great deal of money into building public libraries and Rockefeller did things like setting up who universities. These were clearly not the actions of men who were simply robbers.
Thus, we can say that both titles are apt. These rich businessmen did get their wealth by using cutthroat and ruthless methods. However, they also helped build the American economy, and improved standards of living both through charity and through the creation of processes that allowed cheaper consumer goods to be produced.