I'm sort of stealing the idea from t3suji, but here goes:

If the module is projective, i.e. $PD = 0$, then $FD \leq 1$.

If the module is not projective, i.e. $PD > 0$, then $PD = FD$.

The first statement is t3suji's comment. The second statement follows by first observing that if $F$ is the free module on countably many generators, there is an exact sequence of length $n+1$,

$0\rightarrow F\rightarrow F\rightarrow\ldots\rightarrow F\rightarrow 0$,

so long as $n\geq 1$. Then if we have a projective resolution

$0\rightarrow P_n\rightarrow\ldots\rightarrow P_0\rightarrow M\rightarrow 0$,

we can make the free resolution

$0\rightarrow P_n\oplus F\rightarrow\ldots\rightarrow P_0\oplus F\rightarrow M\rightarrow 0$.