An independent scientist (or gentleman scientist) is someone who pursues scientific research while being independent of a university or government-run research and development body. “Self-funded scientists practiced more commonly from the Renaissance until the late 19th century … before large-scale government and corporate funding was available.” (Wikipedia)
Independent scientists are amateurs in the sense that they are doing scientific research for the love of it (the word is from the French amateur, “one who loves”) rather than as an occupation. They may have an occupation in a related field such as teaching science but their scientific research is done on their own time. Or they may be professional scientists in a specialty other than their research.
I remember years ago hearing the great Hungarian mathematician Paul Erdős remark that an “amateur mathematician” had done work in number theory. He explained that the amateur was a professional mathematician but not a professional number theorist. That made the person an amateur number theorist. It is the same with professionals in any specialty outside their own.
Some great scientists were professors of mathematics, such as Galileo, who was a professor of mathematics at the University of Padua, and Isaac Newton, who held the Lucasian Chair of Mathematics at the University of Cambridge.
In the history of science many breakthroughs have been done by amateurs. Here are some great amateurs or independent scientists:
Albert Einstein – physics
Antonie van Leeuwenhoek – microbiology
Charles Darwin – biology
Gregor Mendel – genetics
Joseph Priestley – chemistry
Michael Faraday – electromagnetism
William Herschel – astronomy
One could add others who were primarily inventors such as Thomas Edison and the Wright brothers, since science is often given credit for inventions.
On a related note, Robert A. Stebbins wrote Amateurs, Professionals, and Serious Leisure (McGill, 1992) and other works on productive uses of one’s free time.