Cosmonaut is the Russian word for a person who travels in space – or, to put it another way, it’s the Russian word for Astronaut. It’s an Anglicization of the Russian word космонавт.
Other than the linguistic source of the word, there is no practical difference between the two words. Dictionary.com defines astronaut as “a person engaged in or trained for spaceflight.”, and defines cosmonaut by referring to the definition for astronaut.
The cosmonaut vs astronaut argument (or should it be the astronaut vs cosmonaut argument) is really now just a historical oddity. Given that astronaut is the definition most commonly used in the United States, the rest of the English speaking world, and most of Europe, it is likely that the term astronaut will win out over cosmonaut in the longer term, possibly even in Russia.
More recently, other foreign words for astronaut have entered the lexicon – for example, Chinese astronauts are sometimes referred to as Taikonauts, and I have seen Indian astronauts, on a couple of occasions, as vyomanauts. The root of all of these words is the local language’s term for space.
Famous Soviet and Russian cosmonauts include Yuri Gagarin, who became the first man in space on April 12, 1961, narrowly beating US astronaut Alan Shepard, who first flew into space on 5 May 1961. Valentina Tereshkova was the first female cosmonaut, whose mission took place in June 1963, almost exactly 20 years before the first female Astronaut from the United States, Sally Ride.
Other notable cosmonauts include Alexei Leonov, the first person, and therefore the first cosmonaut to spacewalk in March 1965, and Valeri Polyakov, who holds the record for the most days in space in one trip – between 8 January 1994 and March 22, 1995 he spent 437 days and 18 hours aboard the Soyuz space station. Added to his early stay of 240 days and 22 hours on Soyuz in 1988/1989, that makes a total of 678 days in space – almost two years.