I have the impression that the names Columbus and Copernicus are not the forms actually used on a daily basis by those people, but rather Latin forms used to identify them in in scholarly or academic connection.
The vast majority of Copernicus's surviving works are in Latin, which in his lifetime was the language of academia in Europe. Latin was also the official language of the Roman Catholic Church and of Poland's royal court, and thus all of Copernicus's correspondence with the Church and with Polish leaders was in Latin… The surname Copernik, Koppernigk is recorded in Kraków from c. 1350, in various spellings… During his childhood, about 1480, the name of his father (and thus of the future astronomer) was recorded in Thorn as Niclas Koppernigk. At Kraków he signed himself, in Latin, Nicolaus Nicolai de Torunia (Nicolaus, son of Nicolaus, of Toruń). At Bologna, in 1496, he registered… as Dominus Nicolaus Kopperlingk de Thorn – IX grosseti. At Padua he signed himself "Nicolaus Copernik", later "Coppernicus". The astronomer thus Latinized his name to Coppernicus, generally with two "p"s (in 23 of 31 documents studied), but later in life he used a single "p". On the title page of De revolutionibus, Rheticus published the name (in the genitive, or possessive, case) as "Nicolai Copernici".
I’ve noticed too that the mathematician Gauss is sometimes called Karl (with a K, that seems more German), and sometimes called Carl (with a C, that seems to associate it with the Latin name Carolus.
How, when and why did Latin cease to be the language of international scholarly communication?
To expand the answer of b.Lorenz with few examples: it was a slow process, and the speed in various areas was very different. Some examples. Proceedings of St. Petersburg Academy (mid 18th century) had the following rule: papers in all sciences are published in French, except mathematics, mathematics was published in Latin.
The last important mathematical book that I know which was written in Latin is Fundamenta Nova by Jacobi (1829). But no one wrote on physics or chemistry in Latin in 19th century.
But most journal papers were not published in Latin since the middle of 19th century, in all sciences.
Latin disappeared from the mandatory curricula in high schools and universities in the beginning of 20th century (in many European countries like Russia and England).
It come about at different times in different subjects, for example Latin names of conditions and anatomical objects are still used in medical science. It generally happened in the 18th and 19th century.
The answer of Alex gives some nice examples on when it happened.
In the Middle Ages the only subject considered worthy for scholarly interest were theology and philosopy and ancient history (this is a stereotypical oversimplification, there were books on medicine, herbs, art, etc... too), so the main occupation of scholars was to read works of ancient auctors writing in Latin. (or rarely Greek, Hebrew or Arabic) After the 17. century new disciplines and fields emerged, many of them based on experiments or newly invented everyday activities (like economics or engineering). It would have been cumbersome to invent, introduce and learn latin worlds for all these new concepts.
The Reformation made it possible to work in native tongue in theology and religious teaching, and after that the use of modern languages slowly permeated all the other areas of written word like law, governance or high literature.
In the most critical decades of the 18th and 19th century it was expected from an educated European man to understand French, so there was a replacement international language ready.
When the international cooperation needed for 'big science' became important in the 20th century, Anglo-saxon countries had already grabbed the lead in many modern sciences and technologies, so their language became the standard.
Latin was considered 'cool' in Renaissance and Classicism, while it was definitely 'uncool' in Romanticism.
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