= Solution
The forcing $\mathbb P=\operatorname{Fn}(\aleph_1^N,2,\aleph_1^N)$ is $\aleph_1^N$-closed, so it adds no reals:
$$
\wp(\mathbb N)\cap N[G]=\wp(\mathbb N)\cap N.
$$
By contrast, $\aleph_1^M$ is countable in $N$. The union of the $\mathbb Q$-generic filter is a total binary function
$$
h:\aleph_1^M\longrightarrow2.
$$
Fix in $N$ a bijection $e:\mathbb N\to\aleph_1^M$. Then $r(n)=h(e(n))$ is a real in $N[H]$. It is not in $N$: for any ground-model real $r_0$ and any $p\in\mathbb Q$, some coordinate of $\aleph_1^M$ is outside $\operatorname{dom}p$, and extending there forces $r$ to differ from $r_0$. Hence $\mathbb Q$ adds a new real, and
$$
\wp(\mathbb N)\cap N[G]\ne\wp(\mathbb N)\cap N[H].
$$
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