If two paths are of the same length, they are equally long. But then, if they're of the same distance, they are equally... distant?
It doesn't sound at all right and I strongly suspect that it's just not a case when a switcharoo between nouns allows for the equivalent on the related adjectives. In my ears, the second version sounds like they are equally distant from each other, without the actual requirement of being equally long. Then again, that statement makes no sense because two objects are always equally distant from each other (physical ones, at least).
Which is it?