In 1492, Columbus [re]discovered America for Spain after being rejected by Portugal. However, less than a decade later in 1499, Italian explorer Amerigo Vespucci was hired by Portugal to explore the New World. Later in 1508, Vespucci became Chief Navigator for Spain. America ultimately gained its name from Vespucci, not Columbus.
Now, I know Columbus' character and methods of colonization and interaction with natives were controversial even in his own time; he also seemed at times an anti-diplomatic character that couldn't get along with Spanish officials.
But it seems a bit strange that after someone who struggled to get any funding for exploration, then made solid gains, would be contemporarily eclipsed by a different, yet-unknown character (Vespucci), who (it seems) rapidly became a new favorite, and despite not being as renowned as an explorer by modern standards (compared to Columbus or Magellan), seemed to have a much easier time of getting known in royal courts, getting expeditions funded, and obtaining lucrative, high-paying posts. Being born in the family of a notary, Vespucci also wasn't particularly high-birth or wealthy, so we can't use that to explain him having seemingly a smoother career than his contemporaries.
So what was it so special about Vespucci, compared to Columbus or Magellan, that seemingly made him a contemporary favorite?
We shouldn't look into these events like some kind of a competition between Columbus and Vespucci because it really wasn't. We should look at it like three separate things, intersecting at certain points: first is the career of Columbus and his fall from grace, second is the career of Amerigo Vespucci and final is the spread of name of America.
Columbus
First to start with Columbus. From 1492 to his death in 1506 he did a total of 4 voyages to the New World, the one in 1492, the next in 1493-1495, third in 1498-1500, and fourth 1502-1504. So, even with the controversy I will now talk about, we can see that Columbus was continuously making voyages until his death. He was never completely discarded, let alone in the name of Vespucci.
The terms of the agreement made before the first voyage granted Columbus rank of Admiral, position as Viceroy of the new discovered lands, tenth of taxes from the territories, and the power that any voyage to the area had to get his approval first. While these rights were in some similar form granted to earlier (portuguese) explorers, it seems soon after the success of the first voyage some factions on the Spanish court, namely bishop Fonesca, started to consider those rights excessive and unfavorable for the crown and look into ways to get rid of Columbus. During the second and third voyages, complaints - real and exaggerated - started pouring which culminated with Columbus' arrest and bringing to Spain in irons in 1500 (a theatrics Columbus himself played up). Subsequent inquiry restored Columbus but barred him from approaching island of Hispaniola (basically the only Spanish settlement by that point) and as mentioned, he made a final voyage in 1502.
Vespucci for Spanish
These events will tie us in with Vespucci. In 1499 Bishop Fonesca, without Columbus permission, organized expeditions to what is now South America but then was still largely unknown. One was lead by Vicente Yáñez Pinzón, the other by one Alonso de Ojeda. It is the Ojeda expedition where Vespucci participated, but as wee see not as overall commander but as one of the captains of several ships.
So, what made Vespucci qualified to be there? We aren't really sure. Vespucci does not seem to had have previous relevant experience in voyages and ships, but on the other hand it was not at all unusual to have the captain of a ship be a nobleman or a financier with no sea experience, with the experienced pilot actually in charge of maritime affairs.
Vespucci was a Florentine representative of Medici bankers, was in Seville - Cadiz for some time before 1499 engaging in trade and very likely involved in funding the expeditions. It is thus assumed he was placed on a expedition as representative of the bakers of the expeditions, a common occurrence. He seems to have had a enjoyable personality (unlike Columbus who antagonized many and was many times described as vain and pompous). And he seems to had exceptionally good background in astronomy, cosmography and cartography as by his own observation he was one of the few on the ships to be able to determine latitude by stars and sun. This too fits in our understanding that in 1480-1500 most of the navigation was done by compass, dead reckoning and charts, with pilots relying on their experience in winds, currents and coasts while still not familiar with celestial navigation. For example Columbus, while considered a great pilot and sailor, seems to had lacked in the astronomy skills, and while he could use a quadrant and North Star, he seems to have had problems with an astrolabe in determining latitudes. So Vespucci's skill might be really valued by his contemporaries.
Vespucci for Portuguese
Soon after Vespucci's return from the above expedition for king of Spain in 1500, he was approached by king of Portugal to lead charting of Brazil coast. Why would King of Portugal choose Vespucci over Columbus, isn't really surprising. Columbus was in service of King of Spain, and had funding, titles and agreements with them which he wouldn't and couldn't break. Why King of Portugal would choose Vespucci over his many other captains actually is perplexing. We do have a theory that he came through recommendation of another Florentine Bartolomeo Marchioni, a banker based in Lisbon who was one of the most important people on Portuguese court. Also Vespucci's previous experience in South America from his Spanish Voyage might be useful, and his sort of neutrality could be a key in the very sensitive issue on determining the borders of Portuguese and Spanish in South America on the basis of Treaty of Tordesillas from 1494. In such a capacity Vespucci charted Brazil.
Name of America
The spread of name of America is a completely separate issue on which books can be and indeed are written. To keep the long story short, after the publication of Columbus letters of his first and second expedition, the educated european public was in want of any new information on the New Lands. In such an atmosphere letters attributed to Vespucci appeared (i touched on the issue of the authenticity in this recent post) in Italy and spread through rest of Europe (with exception of Iberia). One of this letters identified the new lands as a new continent in the southern hemisphere unknown to ancient authors who divided the earth into three landmasses (europe, asia, africa). This prompted a german cartographer Waldseemüller to place a name of America on the southern landmass thinking Vespucci was the first one to discover it. While Waldseemüller himself later backtracked when he was confronted with more information, the name was released "in the wild" and in a slow and convoluted process ended up being universally accepted. But it has to be mentioned here that the Spanish never used the name of America in any form before 18th century.
Pilot Major and Conclusion
When Vespucci was made Pilot Major in 1508, Columbus had already died so it's not like it was favoring one over another. The role of Pilot Major was to train pilots in celestial navigation by the stars and sun(!) to determine latitude as well as keep the universal map of the world called Padrón Real updated with new information. It is a wonder if Columbus if alive would be qualified for, or interested in, this job. Opposite this Vespucci was never given titles of Admiral or Viceroy like Columbus so I would argue Columbus was given more recognition than Vespucci ever was. The same goes for their expeditions. Columbus was commander of his own expeditions, which he proposed and helped organize and was given plenty of rewards for them. Vespucci was a participant in one, and commander of another expedition, both of which were organized by others and on which he was merely appointed. Comparing them does not seem very fair.
Adoption of name of America was it's own process that actually had nothing to do with Spanish or Portuguese, but instead with the rest of Europe who were filling in the blanks from the available material
Basically: Vespucci never "eclipsed" Columbus in the contemporary Spanish court even with Spanish factions actively disliking Columbus. He only became more famous in the rest of Europe through (non authentic) letters that circulated freely