Is module isomorphism problem easier if we have presentations? Let $R = k[x_1,\dotsc,x_n]$ (feel free to restrict $n$ and $k$ if necessary). Assume we work in the graded setting, where everything is graded by $\mathbf{Z}^n$. Let $M, N$ be $R$-modules. In general, it is difficult to decide if $N, M$ are isomorphic (see Brooksbank, Wilson). I wonder if the situation gets easier if we are given free  presentations $F_1 \xrightarrow{f} F_0 \to M$ and $G_1 \xrightarrow{g} G_0 \to N$ of the modules. One could then rephrase the problem as:
Decide if there are homogeneous morphisms $\phi_1\colon F_1 \to G_1, \phi_2\colon F_2$ such that 1. $g \phi_1 = \phi_0 f$ and 2. $\phi_0$ induces an isomorphism $F_0/f(G_1) \to G_0 / g(G_1)$.
Note we can represent morphisms of free modules as matrices with entries in $R$, and actually, (by homogeneity) by matrices with entries in $k$. The first condition is linear in the entries of the $\phi$, so that's easy. The second will certainly not be linear, since the solution space is not.
Still, is there any easy way to answer the question?
Edit: we may assume that the presentations are minimal; I.e., $F_1, G_1$ and $F_0, G_0$ have the same grades of their generators, so $F_1 \cong G_1$ and $F_0 \cong G_0$, for otherwise, the cokernels will not be isomorphic for sure. So the question can be simplified even further:
$\newcommand\coker{\operatorname{coker}}$Given $f, g\colon F_1 \to F_0$, decide if $\coker f \cong \coker g$.
 A: It looks like Brooksbank & Wilson reduce from isomorphism testing in generator relation models; namely, $\textsf{EndoConj}_{k}, \textsf{AlgebraIso}_{k}$. (These may not be exactly the problem you posed, but it should give some intuition that generator-relation models are unlikely to be easy.)
One of Wilson's techniques that appears in a lot of his works is to try to find substructures that behave locally like vector spaces, and then to glue those structures together in compatible ways. This is effectively group cohomology, but a lot of his papers express this in the language of linear algebra. We can usually do linear algebra efficiently.
Given two structures where we have to figure out how to first match up the local vector spaces before trying to glue them together, we are intuitively (and often, precisely) dealing with some flavor of $\textsf{Code Equivalence}$, which is $\textsf{GI}$-hard.
The $\textsf{Module Isomorphism}$ problem in Brooksbank--Luks gives us the generators and asks if they are compatible, whereas Brooksbank & Wilson deal with the problem of deciding if there exist compatible generating sequences.
