The cross-entropy is identical to the KL divergence plus the entropy of the target distribution. The KL divergence equals zero when the two distributions are the same, which seems more intuitive to me than the entropy of the target distribution, which is what the cross-entropy is on a match.
I'm not saying there's more information in one of the other except that a human view may find a zero more intuitive than a positive. Of course, one usually uses an evaluative method to really see how well classification occurs. But is the choice of the cross-entropy over the KL divergence historic?