[Scenario:] You are on a long trans-Atlantic flight from Heathrow Airport (London, UK) to DFW Airport (Dallas-Ft. Worth, USA) seated next to a Brit. He asks you, "I understand that there is often a debate in your country as to whether the industrialists at the end of the 19th century should be called "Robber Barons" or "Captains of Industry," what is your analysis?"

"Robber barons" and "captains of industry" are both terms for the same group of people: a handful of industrialists and entrepreneurs who became extremely wealthy in the late 19th and early 20th centuries in the United States.
What you call them depends on your political orientation. "Robber baron" was the term assigned by the progressive press. This term focuses on the darker side of how these wealthy people amassed their wealth: through violently breaking up attempts of workers to unionize for higher pay and safer working conditions, by creating monopolies of goods or services and then raising prices very high, and by bribing politicians to achieve political ends. Those who focused on these aspects of the very wealthy perceived them as getting ahead by robbing the people who couldn't afford to pay. They saw them as criminal for paying low wages and charging high prices.
Captains of industry was the complimentary term assigned by the conservative, pro-business press. This group focused on the positive aspects of what the wealthy entrepreneurs did for the country: the wealthy created jobs, for example, and also gave generously to charity. Andrew Carnegie, to name just one individual, built many libraries that were open to the public. Those who saw the extremely rich in a positive light argued that they were leaders, risk takers, and role models for other Americans.
I will leave it up to you to explain to your British friend what your opinion is as to whether these entrepreneurs did more evil or good for the United States. One question you might ask yourself is whether you would want to live in the world in which they prospered or a later world in which laws and regulations curtailed some of their power?

"Captains of Industry" and "Robber Barons" are simply two ways of viewing the major industrialists of the late-19th century and early-20th century. This period saw rapid industrialization, urbanization, and exponential developments of many first-world economies, such as the United States and Great Britain.
Additionally, the Industrial Revolution birthed many world-changing inventions, such as the automobile and airplane. Industrialists such as Henry Ford, Andrew Carnegie, and Cornelius Vanderbilt were seen as influential and powerful businessmen at the top of their respective industries.
However, there are critics—mostly those with a socialist, anarchist or communist ideology—who believed that they gained and maintained their wealth through unethical business practices. Although those who call these industrialists "robber barons" come from a certain political ideology, their criticisms had basis in truth.
For instance, Cornelius Vanderbilt, who amassed great wealth from the railroad and shipping industries, was known to receive money from government-subsidized shippers. This allowed Vanderbilt to not compete on their routes. Vanderbilt made his second fortune from the Civil War by supplying steamships, which some historians saw as war profiteering.