$R$ is ring with $a\subset R$ ideal. $M$ is an $R-$module and submodule $N\subset M$.
"We note that, when $N$ is a submodule of $M$, the $a$-adic topology of $N$ may be different from the topology of $N$ as a subspace of $M$."
The theorem following above assertion indicates that if $M$ is a noetherian module, the $a$-adic topology of $N$ coincides with the subspace topology on $N$ when $M$ has the $a$-adic topology.
$\textbf{Q:}$ What is difference between $a-$adic topology and subspace topology and what is the example s.t. $a-$adic topology differs from subspace topology? $a$-adic topology is defined by $a^nM$ as open neighborhood basis of $0$ as in $p$-adic sense.
To see $\textbf{Q'}$'s claim of equivalence when $N$ is finite module over noetherian ring $R$, I denote $N'$ as $N$ in $M'$ $a-$adic subspace topology and $N$ as the $a-$adic topology. It is clear that $Id_N: N\to N'$ is continuous by $a^nN\subset a^nM\cap N$. It suffices to show $Id_N:N'\to N$ is also continuous. From Artin-Rees, I have $\exists r\geq 0$ s.t. for all $n>r$ $a^nM\cap N=a^{n-r}(a^rM\cap N)$. Note that statement holds for all $n>r$. In particular, $n-r\geq 1$ holds as well. Thus we have $a^nM\cap N\subset a^{n-r}N$.(Note that you can choose $n-r$ to start with.) Therefore, this map is also continuous. Thus $Id_N$ is homeomorphism.[This is basically Nagata's proof in the book.]
$\textbf{Q':}$ What is topological interpretation of Artin-Rees lemma? Subspace topology of $N$ as $M$ with $a$-adic topology is exactly $N$ own $a$-adic topology?
Ref. Nagata, Local rings, Chpt 2, Sec 16.
You sorta ask many questions in one, so it is difficult for me to know what exactly you are looking for. However, I hope that the following example would allow you to answer what you need.
Let $M$ be the ring $\Bbb Z[x_1,x_2,\ldots]$ (allowing infinitely many terms in a product, as long as all the powers are finite), and $N=(x_1\cdot x_2\cdot\ldots)M$. Let ${\frak a}=x_1M+x_2M+x_3M+\ldots$. Then the $v_{\frak a}$-adic topology on $M$ acts as $$v_{\frak a}(x_1^{a_1}\cdot x_2^{a_2}\cdot\ldots)=a_1+a_2+\ldots.$$ Then $v_{\frak a}$ is a non-trivial valuation on $M$. However, $v_{\frak a}(y)=\infty$ for all $y\in N$. Therefore, $v_{\frak a}$ reduces to the trivial topology, when viewed as a subspace topology on $N$.
On the other hand, we can still have the $v_{\frak a}$-adic topology on $N$, via $$v_{\frak a}(x_1^{a_1}\cdot x_2^{a_2}\cdot\ldots)=(a_1-1)+(a_2-1)+\ldots.$$
This $v_{\frak a}$-dic topology on $N$ is not trivial, therefore not equal to the subspace topology (which is trivial).
The idea is, if you have a Noetherian ring, then it's not "big enough" to make valuations level out on a non-trivial submodule -- and therefore, such esoteric examples are not possible.