I don't think there is much of a difference in regards to how these two phrases are used in everyday English conversation. However, as to something, unlike why, is actually a phrase on its own and somewhat similar to the expression as per, but probably not exactly the same. It can be used with anything you like. So, you could say:
As to the matter at hand, we must act firmly and without delay.
As to why I was late today, I told him that I got stuck in a one-hour traffic jam on my way to work.
But let's now get back to your question. First of all, your sentence is correct. Secondly, I don't think that in your particular case here there would be any difference whether you use as to or not. But we could turn things around and rephrase my second example in such a way that there is no as to in it:
In answer to the question why I was late today, I told him that I got stuck in a one-hour traffic jam on my way to work.
Just notice how the phrasing has changed compared to the original sentence. So, to sum things up, while why is simply an adverb, as to is always as to SOMETHING where something can be anything, including why.