The assertion is false. Here is how to construct a counterexample.

- Let $A = XX^T + Y + Y^T$ where $Y \ge 0$ (elementwise)
- Let $B = XX^T$

Then, by construction $A$ is a copositive matrix (sum of semidefinite plus symmetric nonnegative matrix), and $B$ is copositive too (because it is semidefinite). Moreover, $A-B$ is also copositive because it is just a symmetric nonnegative symmetric.

However, if you try the above recipe to construct $A$ and $B$, then you get the following counterexample (via Matlab again) very rapidly.

$ X = \begin{pmatrix}
-1.8393& -0.9342\\\\
1.7632 & 1.6479
\end{pmatrix}$

$Y = \begin{pmatrix}
1.9949& 2.0663 \\\\
2.3393& 0.1889
\end{pmatrix}
$

$A = \begin{pmatrix}
8.2456 & -0.3770\\\\
-0.3770& 6.2024
\end{pmatrix}
$

$B =
\begin{pmatrix}
4.2558 &-4.7826\\\\
-4.7826 &5.8247
\end{pmatrix}$

Here, we have $\rho(A) = 8.3130$ and $\rho(B) = 9.8867$.