Why does changing the order of quantifiers in Goldbach's conjecture changes its meaning and truth value?

You can try with a simpler example :

$\forall n \in \mathbb N \exists m \in \mathbb N (n < m)$.

It is clearly true, because for every natural number $n$ we can always find a greater number $m$ : it is enough to choose as $m$ the number $n+1$.

Now, swapping the quantifiers we get :

$\exists m \in \mathbb N \forall n \in \mathbb N (n < m)$

that must be read as : there exists a natural number $m$ that is greater then every number $n$, which is clearly false.

It is standard practice in mathematics to instantiate quantifiers from left to right. Hence the standard way to interpret the second statement is: there is a prime $p$, and a prime $q$, such that for all even $n$, the equation $n=p+q$ holds.

That is, $p,q$ are chosen before $n$ is instantiated, so the statement must hold without changing either $p$ or $q$, for every $n$. This is absurd, and false.

In English we often rely on context to disambiguate sentences, so we are used to interpreting them "however makes sense"; however in mathematics we want extra precision, including the ability to make statements that are absurd if we so desire.