Which tessellation of the sphere yields a constant density of vertices?
There are several possible ways of defining "density on a sphere", each one giving somewhat different results.
Alas, most of them have some "maximum number of vertices" that give exactly equal density. Above that maximum number, further tessellation can at best approximate constant density. (That approximation is more than adequate for many purposes).
"Unfortunately, it is a well-known group theoretical result that there are no completely regular point distributions on the sphere for N > 20." -- Max Tegmark
Equal density as equal areas of the triangles formed by the vertices: You can tessellate a sphere to give a geodesic sphere such that every triangle has exactly equal area, to any desired resolution, using any equal-area projection such as the Snyder equal area projection. (A few people use geodesic grids based on this principle).
Equal density as congruent triangles formed by the vertices: When a person builds a geodesic dome out of panels, it would be super-convenient if every panel were identically the same size and shape. Alas, the maximum "size" is the 120 identical faces of the hexakis icosahedron (aka disdyakis triacontahedron). Any convex solid with more than 120 faces must necessarily have 2 or more kinds of faces.
Equal density as minimum-energy configurations of charged particles: Min-Energy Configurations of Electrons On A Sphere. You can put any integer number of repelling particles on a sphere, and calculate some minimum-energy configuration.
Equal density as equal distance from every vertex to the N nearby vertices: When a person builds a geodesic dome out of struts, it would be super-convenient if all the struts were the same length. Most "naive" methods of dividing the large triangles of the icosahedron into smaller triangles generates lots of different edge lengths; but there are ways to "tweak" the tessellation subdivision in order to minimize the number of different lengths of edges. (Fewer unique lengths requires fewer jigs in manufacturing and fewer spares needed to replace any damaged strut). Alas, the maximum "size" of a strictly convex polyhedron made entirely of equilateral triangles (convex deltahedron) is the 30 edges of the icosahedron. (You could try to make the pentakis dodecahedron out of 60 equilateral triangles, giving 90 equal-length edges, but then it would be slightly concave). Any strictly convex solid made of triangles with more than 30 edges must necessarily have 2 or more lengths of edges.
A few more notes on sphere approximation.
See the links below. They're not about tesselation, but about evenly distributing points on a sphere (from which you can extract a tesselation):
Distributing Points on a Sphere
Distributing many points on a sphere
Tables of Spherical Codes with Icosahedral Symmetry
Recursive Zonal Equal Area Sphere Partitioning Toolbox
Frequently-Asked Questions About Spheres
Evenly distributing n points on a sphere