Map projections are important because it is the main way that we familiarize ourselves with how the world looks. Most people do not use 3-D projections of the world when referencing distance or area, so it is important to understand how different maps are distorted so one doesn't believe that what they are looking at is exactly how the earth looks.
Conformal projections like the Stereographic or Mercator projection are nice because they map the earth on a grid which makes allows us to make UTM maps and is easier to split into segments. However, they also heavily distort some areas and also can alter the actual distance between two locations.
Equidistant projections are very useful for distance related subjects, which apply to a wide range of people. Governments may use equidistant maps for missile defense or people may use equidistant maps when figuring out how far they want to hike. The downside with equidistant maps is that they are also distorted, especially the area. In the Plate Carree projections, Alaska looks to be the size of Brazil, and in the Sinusoidal projection, Alaska is barely visible. In you grew up only looking at these two projections, you would either think Alaska was much bigger than it was or be unaware of Alaska.
Equal area projections are good for an overall picture about the relative size of objects. The Mollweide and Bonne projection are not projected too oddly (in comparison to some of the other equal area projections). However, these maps are not easy to manipulate to form something like quadrangles or very useful in measuring distance.
Any projection alone is not sufficient for portraying everything that needs to be, however, certain maps can serve certain purposes, and a combination of maps can fulfill all of our needs for accuracy and convenience.
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