What is the difference between the Yang--Baxter equation and the quantum Yang--Baxter equation? For a vector space $V$ and a linear operator $R:V \otimes V \to V \otimes V$, we say that $R$ satisfies the Yang--Baxter equation if
$$(R\otimes id)(id\otimes R)(R\otimes id)
 = (id\otimes R)(R\otimes id)(id\otimes R).$$
If instead $R$ satisfies
$$R_{12}R_{13}R_{23} = R_{23}R_{13}R_{12}$$
we say that $R$ satisfies the quantum Yang--Baxter equation.
So what is the difference between the Yang--Baxter equation and the quantum Yang--Baxter equation? I guess that YBE came first and then came QYBE, but I don't see what is quantum about QYBE. What different properties do both have, and why should one consider them two versions of the same thing?
 A: Your two equations are equivalent, and are both versions of the quantum YBE. (The question from the comments does a good job of answering your classical versus quantum question.)
Write the first as
$$
(c \otimes id)(id \otimes c)(c \otimes id) = (id \otimes c)(c \otimes id)(c \otimes id)
$$
and the second as
$$
R_{12}R_{13}R_{23} = R_{23}R_{13}R_{12}.
$$
Let $\tau : V \otimes V \to V \otimes V$ be the permutation map $\tau(v_1 \otimes v_2) = v_2 \otimes v_1$.
Then a solution $R$ to the second gives a solution $c = \tau R$ to the first, and vice-versa.
Whether or not to include the permutation in the definition of $R$-matrix is just a convention.
In the quantum groups literature, $V$ is a module for a quasitriangular Hopf algebra $H$ and $R$ can be given by the action of the universal $R$-matrix $\mathcal{R} \in H \otimes H$. Since $\tau$ will not be given by the action of something in $H \otimes H$, it's natural to leave it out.
(I think the YBE for $R$ is also more natural in the spin chain context, but I know less about that.)
However, in quantum topology we focus more on $c = \tau R$ (because it gives braid group representations) and less on $R$, so some authors eliminate the distinction and call $c$ an $R$-matrix. I prefer to call $c$ a braiding and $R$ an $R$-matrix, but this convention is not universal.
