There is no such regular polyhedron whose volume is equal to the difference between the volumes of its circumsphere and its insphere

The quotients of length and distance data describing all sorts of regular polyhedra, incl. the Archimedean ones, etc., are algebraic numbers. It follows that the volume of such a polyhedron is an algebraic multiple of $\rho^3$, where $\rho$ denotes the inradius of $P$. Similarly the circumradius $R$ of $P$ is an algebraic multiple of $\rho$. But the difference between the volumes of the circumsphere and the insphere of $P$ is given by $${4\pi\over3}(R^3-\rho^3)=\pi\>\lambda \rho^3$$ with $\lambda$ algebraic, and is not of this kind.

Not as elegant as Christian Blatter's answer, but here is a Mathematica code to compare the values given in wikipedia's table.

comp[r_, V_, R_] := N@{4/3*Pi*R^3 - 4/3*Pi*r^3, V}
comp[1/Sqrt[6], Sqrt[8]/3, Sqrt[3/2]]
comp[1, 8, Sqrt[3]]
comp[Sqrt[2/3], Sqrt[128]/3, Sqrt[2]]
comp[phi^2/xi, 20 phi^3/xi^2, Sqrt[3]*phi]
comp[phi^2/Sqrt[3], 20 phi^2/3, xi*phi]

{7.41029, 0.942809}
{17.5768, 8.}
{9.5676, 3.77124}
{45.9338, 61.305}
{14.3614, 17.4536}

The three first ones have a volume smaller than the difference between the circum- and in-spheres volumes, and that is the reverse for the two last ones.