The two map projections that are conformal are the Mercator map and the North America Lambert Conformal Conic map. The distance between Washington, D.C. and Kabul, Afghanistan in the Mercator map is about 10,168 miles. The distance between the two cities in the North America Lambert Conformal Conic map is about 6,875 miles. The two map projections that are equal area are the Cylindrical Equal Area map and the USA Contiguous Albers Equal Area Conic map. The distance between Washington city and Kabul city in the Cylindrical Equal Area map is approximately 10,113 miles, while the distance between these two cities in the USA Contiguous Albers Equal Area Conic map is about 7,320 miles. The two map projections that are equidistant are the USA Contiguous Equidistant Conic map and the North America Equidistant Conic map. The distance between the cities of Washington and Kabul in the USA Contiguous Equidistant Conic map is about 7,278 miles while it is approximately 6,951 miles between these cities in the North America Equidistant Conic map.
Map projections are important because they allow maps to represent the Earth accurately. Maps need coordinates to give specific locations to areas on the surface of the Earth and can be given in different coordinate systems. There are geographic coordinate systems which are degrees of longitude and latitude. These can be represented in degrees, minutes, and seconds, or in decimal degrees. There are also projected coordinate systems from the projection of maps. A projection of a map is like putting the 3-D world onto a 2-D map. A datum, which is a 3-D frame of reference or model of the Earth, needs to be selected first and is used to define surface locations. It helps define the origin and orientation of points on the surface of the Earth. Then the ellipsoid that was fitted to the geoid of the Earth (where all locations on the Earth are leveled to it so it is a gravity model with an equi-potential surface) is mathematically converted and transformed to the flat plane of a map.
There are many types of map projections. Different types of projections preserve different properties onto maps. For example, local conformal map projections preserve local shapes and angles. The parallels and meridians intersect at right angles. This can be seen in the Mercator map and the North America Lambert Conformal Conic map, where the parallels and the meridians intersect at right angles. Equidistant map projections make the distance from the center of the projection to any other place on the map uniform in all directions. This is why the USA Contiguous Equidistant Conic map and the North America Equidistant Conic map have the north pole as the center of the projection and show how areas in all directions from this center are equidistant. Equal area projections make the areas on the map maintain the same proportional relationship to the areas on the Earth that they represent. This is why the continents in the USA Contiguous Albers Equal Area Conic map and the Cylindrical Equal Area map maintain about the same proportional relationship (in terms of area and size) to the actual continents on the surface of the Earth.
Map projections are useful but have distortions. There are discrepancies between map projections and this is the reason why the six different maps have a different distances between the two cities of Washington, D.C. and Kabul, Afghanistan. For example, the North America Lambert Conformal Conic projection preserves direction but distorts distance and area on the map of places on the surface of the earth. This is why the North America Lambert Conformal Conic map shows the continent of South America as being significantly bigger than the continent of North America, which is incorrect. Another example is how the Mercator map projection preserves the shape and direction of places on the Earth on maps but distorts areas on the surface of the Earth on maps. This is why the shape of the continents on the Mercator map have the right shape but are represented with the wrong area. However, map projections can help people make measurements and allow them to compare areas, shapes, distances, and directions of features on maps. Map projections are not very important when a map covers a small part of the Earth's surface (for example, street maps) because distortion is negligible at this scale. Projections are also not important when people are only interested in the relative location of features on a map.