People caught on soon enough that the "Asia" that Columbus had found was not really Asia, but explorers believed they were on the right track and could still reach Asia by sailing around the Atlantic. They began searching for a "Northwest Passage," a river or sea that could get them around the American continent.
They wanted to find the Northwest Passage for the same reasons they wanted to trade with Asia. 1) Exlusive Asian goods such as gold, silk, and spices fetched high prices in Europe, and anyone who could capitalize on that would be rich.
2) Reaching Asia by land was not optimal because of ongoing wars with Turks and the high amount of bandits, and sea travel was faster. As a matter of fact, a lot of money was needed to finance the war with the Turks in the first place.
3) They also wanted to spread Christianity around the world so they could gain holy brownie points.
