I'm going to assume that you're referring to the map that's available at this link; the only "Columbus Map" that I've found a reference to is one which depicts only Europe and part of Africa, making it unsuitable for a comparison to a modern world map. The attached map was created by Heinrich Hammer in 1489, but was supposedly based on a slightly older one created by Christopher Columbus's brother. Mapmaking at the time was not as diligent or scientific as it is today; it sometimes included poor or optimistic mathematical calculations as well as inference from, and deference to, ancient Greek and Roman texts which reported certain landmasses to be in certain areas, regardless of whether they had actually been observed.
Some of the most glaring differences include;
- Scale: The Hammer map appears to put Europe at a fairly reasonable scale to itself, but the farther away things are from Europe, the more distorted they are, particularly East Asia and South Africa.
- Projection: The Hammer map doesn't appear to use any sort of projection system, such as the Mercator, that modern maps use to attempt to compensate for displaying a 3d spherical area on a 2d medium. Neither are there any lines of latitude or longitude.
- Details: Country borders, which typically appear on modern maps, are omitted. Rivers appear to take considerably greater prominence. An attempt is made to include topography (the yellow wiggles are mountain ranges) as well as other land features (trees and cities). Very few islands are named.
- Coastlines: many appear jagged and ribbed, like the teeth of a saw or torn pieces of paper. These features, if present on the actual coastline (they aren't) are not visible on a continental scale.