Say, Q(x,y) is a statement "x has sent a message to y" where the domain for both x and y consists of all students in my class.
Is ∀x∃yQ(x,y) the same as ∃y∀xQ(x,y) ?
The translation for the former one should be:
"Every student in my class has sent a message to some student in my class"
And as for the second part:
"There is some student in my class who has been sent a message by all the students in my class"
They're not the same. Imagine your students formed a circle and each sent a single message to the person on their right. The first is satisfied and the second is not.