We define a random variable to be a measurable function $X$ from a probability space $(\Omega,\mathcal{F},P)$ to a measurable space $(S,\mathcal{B})$. Given a probability measure $\mu$ on $S$ we say that $X$ follows the probability law $\mu$ if the pushfoward of $P$ via $X$ is equal to $\mu$ $$X\sim\mu\text{ if and only if. } X_*(P)=\mu.$$ At this point it seems to me that the probability space $\Omega$ is forgotten, and all reasoning are done thinking to their law.
So my question is: Are there situations where it is relevant to distinguish random variables from their law? If it is just a matter of notation, are there problems that are dealt with significantly better in one formalism with respect to the other?
There is a lot that can be said about a random variable $$ X: (\Omega,\mathcal{F},P) \to (S,\mathcal{B}), $$ just by looking at $\mu:=X_*(P)$. For example, if $S=\mathbb R$, then the expectation $\mathbb E(X)$ may be computed in terms of $\mu $ as follows, thanks to the change of variable formula applied to $x = X(ω)$: $$ \mathbb E(X) = \int_\Omega X(\omega ) \, dP(\omega ) = \int_{\mathbb R} x \, d\mu (x). $$ Similar facts hold for the standard deviation and most other important concepts referring to a SINGLE random variable.
However, statistical notions that depend of TWO random variables, say $$ X_1, X_2: \Omega \to \mathbb R, $$ often cannot be expressed in terms of the probability laws $\mu_1:={X_1}_*(P)$ and $\mu_2:={X_2}_*(P)$, an example being the covariance $$ \text{cov}(X_1,X_2) = \int_\Omega \big (X_1(\omega )-\mathbb E(X_1)\big )(X_2(\omega )-\mathbb E(X_2)\big ) \, dP(\omega ). $$
A clever way to study two random variables $X_1$ and $X_2$, as above, without too much emphasis on the space $\Omega $, is to consider the function $$ T:\omega \in \Omega \mapsto (X_1(\omega ),X_2(\omega ))\in \mathbb R^2, $$ together with the two projections $$ \pi _i:(x_1,x_2)\in \mathbb R^2 \mapsto x_i \in \mathbb R. $$
Besides the fact that $X_i=\pi _i\circ T$, notice that if $Q$ is the probability measure on $\mathbb R^2$ given by $$ Q=T_*(P), $$ we have that $$ {\pi _i}_*(Q) = {\pi _i}_*(T_*(P)) = (\pi _i\circ T)_*(P) = {X_i}_*(P)=\mu _i. $$
This said we may safely replace $(\Omega ,P)$ by $(\mathbb R^2,Q)$, $$ \matrix{ && \kern30pt\mathbb R \cr & & \quad \nearrow_{π_1} \cr Ω & \buildrel T \over \longrightarrow & \mathbb R^2\kern 20pt \cr & & \quad \searrow^{π_2} \cr && \kern30pt\mathbb R } $$ while replacing $X_i$ by $\pi _i$, in the sense that any statistical information one could ask about $X_1$ and $X_2$ can be easily computed in terms of the new random variables $\pi _1$ and $\pi _2$ by means of the change of variables $(x_1,x_2)=T(\omega )$.
Regarding expectation, for example, we have $$ \mathbb E(X_i) = \int_\Omega X_i(\omega ) \, dP(\omega ) = \int_{\mathbb R^2} x_i\, dQ(x_1, x_2) = \mathbb E(\pi _i). $$
Speaking of covariance, we have $$ \text{cov}(X_1,X_2) = \int_\Omega \big (X_1(\omega )-\mathbb E(X_1)\big )(X_2(\omega )-\mathbb E(X_2)\big ) \, dP(\omega ) = $$ $$ = \int_{\mathbb R^2} \big (x_1-\mathbb E(\pi _1)\big )(x_2-\mathbb E(\pi _2)\big ) \, dQ(x_1, x_2) = \text{cov}(\pi _1,\pi _2). $$
Finally, recall that $X_1$ and $X_2$ are said to be independent if $$ P\big (\{\omega \in \Omega : X_1(\omega )\in A,\ X_2(\omega )\in B\}\big ) = $$ $$ = P\big (\{\omega \in \Omega : X_1(\omega )\in A\}\big )\ P\big (\{\omega \in \Omega : X_2(\omega )\in B\}\big ), \tag 1 $$ for any two Borel sets $A,B\subseteq \mathbb R$. Notice that the LHS above is precisely equal to $$ P\big (T^{-1}(A\times B)\big ) = T_*(P)(A\times B) = Q(A\times B), $$ while the RHS is $$ P\big (X_1^{-1}(A)\big ) \ P\big (X_2^{-1}(B)\big ) = {X_1}_*(P)(A)\ {X_2}_*(P)(B) = \mu _1(A)\ \mu _1(B). $$
Therefore (1) may be expressed as $$ Q(A\times B) = \mu _1(A)\ \mu _1(B), $$ which precisely says that $Q$ is the product measure of $\mu _1$ and $\mu _2$.