Is the order of universal/existential quantifiers important?

If you have a formula with existential quantifiers, it is important in which order they appear.

Just to make an easy example:

$\forall$ man $\exists$ woman: the woman is the true love of the man

which is obviously a different statement than

$\exists$ woman $\forall$ man: the women is the true love of the man

The first one means that there a many women - eventually for every man another woman. The second statement means there is (at least) one women that is loved by all men. Good for the woman, eventually bad for the men.

If you have two existential quantifiers or two universal quantifiers, does the order make a difference?

Any number of successive quantifiers of the same kind can be replaced by a single quantifier by combining the quantified variables into a tuple; e.g. $\forall x\forall y$ is equivalent to $\forall(x,y)$. The order in the tuple is irrelevant, and thus so is the order of the quantifiers.