= Solution
The same <cone on an infinite coinfinite ground set> works:
$$
\boxed{E=[2\mathbb N]^\omega.}
$$
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 $*$-<closure> therefore has nonempty $*$-<interior>, so $E$ is \b[not $*$-nowhere dense].
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