"What the difference between a free variable and the usage of $∃x$ ? I mean, isn't $∃x$ the same as some random variable?"
So the issue is:
"What are free variables actually useful for ?"
I think that it may be usefulto start from:
"What are quantifiers useful for ?"
Consider an example from elementary algebra :
$(x + y)^2 = x^2 + 2xy + y^2$.
Usually, we do not use quantifiers to state this "law", but it is implicitly universally quantified; i.e. tha "intended" meaning of the law is :
for every two numbers $x$ and $y$, the above identity holds; i.e.
$\forall x \ \forall y \ [(x + y)^2 = x^2 + 2xy + y^2]$.
Thus, we may think that, in the usual mathematical practice, we may dispense with the universal quantifier.
But we can consider a different algebraic example:
In this case we are not asserting that the equation holds for all values of $x$.
When we ask for the solutions (if any) of equations, we are asking if the existential quantification of the equation holds, i.e. if
$\exists x \ (x^2-2x +1=0)$.
The conclusion is that free variables in "usual" mathematical contexts can be treated as implicitly universally quantified.
In other words, it is not possible to express all the "interesting" facts we want to express simply with universal quantifiers upfront.
"$\exists x$ is the same as some random variable",
we have that the existential quantifier is needed in order to express the fact that in a certain "domain of discourse" there are some objects (at least one) which satisfy a certain property.
For example, we have that in the domain $\mathbb N$ of natural numebrs, there is an object (a number) which satisfy the identity : $x = 0$. Precisely, the number $0$.
Thus, we are licensed to state the following true fact about $\mathbb N$ :
$\exists x \ (x = 0)$.
Of course, bound variables can be replaced without changing the meaning of the formula; thus we can write: $\exists y \ (y = 0)$, and the two formulae express the same fact about $\mathbb N$.
But we have to take care of "interactions" between quantifiers; we cannot start from $\exists x \ (x = 0)$, "existentially instantiate" it (incorrectly) to derive e.g. $(y = 0)$ and then apply UG to conclude with: $\forall y \ (y = 0)$, that is clearly false.
This kind of details are formalized with the rules of the proof system, and different "styles" of proof system can differ on the details, but all must formalize our "intuitive" understanding of what typs of inference are "correct".
Peter Smith's answer above gives a beautiful explanation of the role of free variables in first-order logic proofs.