LOCC is a setting which asks about, for example, transforming $|\psi\rangle$ to $|\phi\rangle$, and what is the maximum probability with which that is achieved.
If you restrict to demanding perfect transformations in both directions, then it turns out that this boils down to local unitary equivalence. You can roughly understand this by realising that if you were to perform a measurement, you remove entanglement from the system, and you can never add that entanglement back in again under LOCC. Thus, including a measurement is not reversible.
So, yes, the CC part is irrelevant in the circumstance where you want a reversible transformation. But once you get to the non-reversible regime (even in the case where you want a transformation with certainty), it's very important because that allows you to make measurements, and send the results to the other person, who can take different actions depending on the outcome. For instance, imagine you have the state
$$
|\psi\rangle=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)
$$
and you want to transform it, under LOCC, to
$$
\alpha|00\rangle+\beta|11\rangle
$$
with certainty. You can do this by Alice performing a measurement comprising two (non-projective) measurement elements
$$
M_0=\left(\begin{array}{cc} \alpha & 0 \\ 0 & \beta \end{array}\right),\qquad M_1=\left(\begin{array}{cc} \beta & 0 \\ 0 & \alpha \end{array}\right).
$$
Once Alice has measured, Alice and Bob either share the state
$$
\alpha|00\rangle+\beta|11\rangle,\qquad \alpha|11\rangle+\beta|00\rangle.
$$
Once Alice has used the CC to tell Bob the result, they can either both do nothing or both apply $X$ so that, in either case, the output is the desired state.
