We have the following classical result:
If $n>m$, then the 1-spheres $S^n$ and $S^m$ are not homeomorphic.
Can we prove that with the Jordan-Brouwer theorem? In particular, we set $S^m= (S^m, 0,...,0)\subset S^n$ and note that the set $\mathbb{R^{n+1}}\setminus S^n$ has two connected components but the set $\mathbb{R^{n+1}}\setminus S^m$ has only one connected components. Thus, $S^n$ and $S^m$ cannot be homeomorphic.
Yes, this argument works. Jordan-Brouwer states that any homeomorph of $S^n$ in $\Bbb R^{n+1}$ separates it. There is, as you say, a homeomorph $S^m$ ($m<n$) in $\Bbb R^{n+1}$ which does not separate it. So $S^m$ is not homeomorphic to $S^n$.