 Solution

The same cone on an infinite coinfinite ground set works:It is $\tau$-closed: a set outside $E$ has an odd point, and the ordinary neighbourhood given by its initial segment through that point remains outside $E$. Its $\tau$-interior is empty, since every ordinary basic open set admits an extension having an odd point. Hence its ordinary closure equals itself and has empty interior, proving that it is a $\tau$-nowhere dense set.

In the Ellentuck topology, however, $E$ is itself a nonempty open set. Its $\ast$-closure therefore has nonempty $\ast$-interior, so $E$ is not $\ast$-nowhere dense.