In this paper, we investigate whether the symbolic and ordinary powers of a binomial edge ideal $J_{G}$ are equal. We show that the equality $J_{G}^{t}=J_{G}^{(t)}$ holds for every $t \geq 1$ when $|Ass(J_{G})|=2$. Moreover, if $G$ is a caterpillar tree, then one has the same equality. Finally, we characterize the generalized caterpillar graphs which the equality of symbolic and ordinary powers of $J_{G}$ occurs.