Intuitive Reason that Quantifier Order Matters
Take $P(x, y)$ to mean $y$ is a parent of $x$.
Then $\forall x \exists y P(x, y)$ means everybody has a parent, while $\exists y \forall x P(x, y)$ means there is someone who is the parent of every son and daughter.
Let $P(O,C)$ mean car $C$ is owned by owner $O$.
Then $\forall C \ \exists O \ P(O,C)$ means every car has an owner.
However, $\exists O \ \forall C \ P(O,C)$ means that some owner owns all cars.
Clearly these mean different things.
Take $P(x,y)$ to be "$x$ is friends with $y$".
One statement says 'everyone has a friend', the other says 'someone is friends with everyone'.