Columbus was not sure how far he had to go. Polar explorers depended on navigational data for survival, as well as success. Travelers on land need somewhat different information from those at sea. The globe is the best way to show the relative positions of places, but a globe that can fit in a ship's cabin cannot show the detail needed for navigation.
Flat maps distort the placement of features, but can show great detail and are portable.
The transformation of map information from a sphere to a flat sheet can be accomplished in many ways, called projections. Mapmakers have invented projections that show distances, directions, shapes, or areas as they are on a globe, at least partially. Different projections have different advantages and disadvantages. Orthographic projections, for example, show shapes as they appear when the globe is viewed from space. Equal-area projections do not distort the size of areas but do distort their shapes.