They seem to show so much, in the relative placement/distances/angles of nodes and edges. But when you get right down to it, almost all that eye-catching detail is random noise. The same "spokes" could be in any radial order. The relative above/below/right/left positioning could be reversed with no loss of meaning. Often dragging a cluster can result in a completely different set of 'nearnesses'. And as andymangold mentions elsewhere in this thread, for WikiWeb even the set of nodes that are expanded from any origin are randomly chosen... so you can't even follow the same path twice.
So these graphs tempt with their visual connotation that they are information-dense and stable like a real map, but then turn out to be splatter-art, pretty but with most of the ink being random noise.
These problems might be fixable with extra layout constraints. What if shorter articles were always up and longer ones down? More-inlinked to the left and less-inlinked to the right? What if edge lengths or thicknesses were correlated to other notions of bidirectional similarity? Of course doing this, in an automated fashion that continues to look nice and meaningful in every corner of the dataset, is quite hard.
Something forcing a little more of a 'tree' feel, at least when moving in certain directions, could help pack more deterministic meaning and text into a small area. (Think vague intimations of Miller Columns within a rendered graph.)