There is a little difference between formula and sentence in the first order logic. Sentence is just a formula without a free variables. I consider where there is a important difference between them. I cannot see any one, for example:
$$ \forall x f(x) = 3 \\ \exists x f(x) = 3 \\ f(x) = 3$$ Only the last one is formula but, without a valuation function it is pointless.