His first voyage in 1497 is the most controversial. If it occurred as described in a certain group of letters, then Vespucci saw the mainland of the New World more than a year before Columbus. However the letter, written to a man in Florence, and signed by Vespucci is probably not genuine. He probably did go to the New World about that time, as a crew member on one of the many less known voyages that flooded out from Spain to the Caribbean Islands after Columbus came back from his first voyage. By 1497 Columbus had already made two voyages to the New World. However, though he firmly believed he was exploring Asia, he was, in fact, only exploring the Caribbean Islands. On his third voyage, Columbus did land on the South American continent, in what is now called Venezuela during August of 1498. In the end, who touched a continent first may not be that important. After all, geographers consider the Caribbean Islands, which we also call the West Indies, part of North America.
Even after his fourth and last voyage, when he realized that South America was a separate continent, Columbus still thought that Hispaniola and the other Caribbean islands were part of Asia. His brother Bartholomew drew a map in 1503 to show these ideas. However Vespucci suspected, even before ever sailed to the New World, that Columbus was wrong. He had underestimated the diameter of the Earth and had grossly over estimated how far east the Asian continent reached. Columbus thought that Cipango (Japan) was about 2,400 nautical miles from the Canary Islands, the starting point of his first voyage. So when he sailed about that far and found the Caribbean Islands he thought he was in Asia. That is why he called the islands the West Indies and the people he found there Indians.
Vespucci suspected Columbus had not gone far enough to get to Asia. So what he had found must be unknown or new lands. When Vespucci got to South America and observed how different the plants and animals there were from those in Asia, he became convinced that, indeed, Columbus had found lands totally separate from Asia.
Here's another question:
By 1503 both Vespucci and Columbus agreed that...
the Caribbean Islands were part of Asia.
South America was a continent separate from Asia.
the distance from the Canary Islands to Japan was about 2,400 nautical miles.