In A Tour Through Mathematical Logic, Wolf states that
Every formula of a first-order language is a statement in the sense of Section 1.2, but not conversely.
In Section 1.2, Propositional Logic, he mentions the following for statements:
Let us use the word statement to mean any declarative sentence (including mathematical ones such as equations) that is true or false or could become true or false in the presence of additional information.
Question: When can a statement not be a formula?
$P(x)$ is a formula, but not a statement.
Formulas can have free variables, and indeed $P(x)$ has a free (unquantified) variable $x$. Because it is not quantified, we cannot assign a truth-value to it, and so it is not a statement.
Indeed, note that I can quantifiy the $x$ either existentially or universally (or, if I had different quantifiers, differently yet). But, $\exists x \ P(x)$ is clearly a different statement than $\forall x \ P(x)$: I can easily come up with a domain and interpretation of $P(x)$ where $\exists x \ P(x)$ is true but $\forall x \ P(x)$ is false. So, I can indeed not assign a truth-value just to $P(x)$, so it is not a statement.
$P(x)$ is a formula though: it does follow the syntactical rules (the 'grammar' if you want) for expressions in FOL. I don;t have the text, but presumably the author lays out exactly those rules in defining formulas, and you can check for yourself that $P(x)$ does follow that definition. We sometimes therefore call formulas 'well-formed formulas' or wff's for short.
Something like $)xP)((,)$ is clearly not a formula ... that's just a string of symbols, but otherwise gibberish