Question: My intuition says there must be difference because $\mathbb Z\subset\mathbb Q$ so there are elements in $\mathbb Q$ but not in $\mathbb Z$.

My question is about whether the following explanation true or not.

There is bijective mapping between $\mathbb Z$ and $\mathbb Q$ (because both are countable) so they are isomorph one another because of it their elements (elements of $\mathbb Z$ and of $\mathbb Q$ ) are just symbol so they are actually the same set with different symbol.

Answer: 
In the category of sets where the morphisms are arbitrary functions, isomorphisms are bijective functions. Since there exist bijections between the integers $\mathbb{Z}$ and $\mathbb{Q}$, they are isomorphic *in the category of sets*. Thus if the only structure you care about is the structure of a set, then you can treat $\mathbb{Z}$ and $\mathbb{Q}$ as the same object. In this setting, one can reasonably say that they are "equal".

However, we don't generally work in the category of sets. Typically, we want more structure than that. For example, in the *category of abelian groups*, the morphisms are group homomorphisms, and isomorphisms are group isomorphisms. A group isomorphism must be bijective, and preserve the group operation, i.e. it is a map $\varphi$ such that $\varphi(x+y) = \varphi(x) + \varphi(y)$ for all $x$ and $y$ in the domain.

In this setting, $\mathbb{Z}$ and $\mathbb{Q}$ are quite different, as there is no bijective map $\varphi : \mathbb{Z} \to \mathbb{Q}$ such that $\varphi(m + n) = \varphi(m) + \varphi(n)$ for all integers $m$ and $n$. To see this, suppose that $\varphi$ is a homomorphism and note that as soon as you know what $\varphi(1)$ is, you have completely determined what $\varphi$ does. If $\varphi(1) = r$, then $\varphi(\mathbb{Z})$ is the set of all integer multiples of $r$ in $\mathbb{Q}$. But $\frac{r}{2}$ is a rational number, and is not in the image of $\mathbb{Z}$, hence it is impossible for $\varphi$ to be surjective onto $\mathbb{Q}$.
Everything you said up until "so they are actually the same set with different symbol" was correct.

They are *isomorphic* in the category of sets (meaning there is a bijection between them), but that does not mean that they are the same set with different symbols. There are other structures on these sets besides the underlying set of elements which distinguish them.

For example, if you want a bijection $f: \mathbb Q \to \mathbb Z$ which also satisfies the property that it preserves addition, i.e., $f(x+y)=f(x)+f(y)$, then you will not be able to find one.