Projective syzygy vs. free syzygy When referring to syzygies, some books refer to free resolution and some books refer to projective resolution. Are they equivalent in some sense? Is it true, for instance, that the $n$-th syzygy in a finite projective resolution is the $n$-th syzygy in a finite free resolution?

No.

A module can have different projective and free dimensions. Very nice examples can be found here. The free dimension of a projective module can be as big as one.

If the ring satisfies some hypotheses, then the answer becomes yes. For example, over a local commutative ring, or a non-negatively graded connected ring (and you are talking about graded modules and homogeneous maps)