There are some very nice comments from /u/Georgy_K_Zhukov and /u/TywinDeVillena in this thread from last week. The basic answer is that Enlightenment figures developed these ideas as pro-science/anti-Church propaganda, and they were taken up with zeal in the 19th-century as a way of promoting the authority of science over other authorities (political, religious, etc.). They are today put out by people who have similar ideologies ("scientism" is what critics tend to call this, and most historians of science are critical of this approach).

Not many people thought the world was flat at the time of either of these people. The idea of Earth as a globe had been around and popularized in bronze-age Greece. Construction and building placement in several ancient Neolithic societies proves a relatively advanced knowledge of the Earth's movement through the Heavens. The Church knew this as well but chose to famously demonize the heliocentric Earth and those who claimed it. They did not deny that the world was a globe.