Richard systematically walks through each type of map and points out the strengths and weaknesses. He has some interesting ideas about improving tile maps, such as the following equal-area but different-orientation tile map:
Several of my prior posts are related to this discussion:
I talk about tile maps here, and point out its built-in every-state-is-equal assumption, which is appropriate only if that assumption holds. Tile map is the newly popular name for equal-area cartograms, a term that Richard uses.
Years ago, I found a case of mis-use of the tile maps.
Recently, I wrote about the invariance property of some choropleths.