Question

Because map projections must turn a 3-D world into a 2-D map, they will always have the problem

Answer 1

Because map projections must turn a 3-D world into a 2-D map, they will always have the problem of: distortion
Converting maps from 3-D into 2-D will often cause some sort of misshape of the map, even though the distance and the size projection between the maps maybe accurate