The differences are the colour definitions in both the latex-image-preamble and asymptote-preamble. In the print version, these are grey scale.
I know you've mentioned in the past that there's a way to rewrite those definitions in the LaTeX output after the fact, rather than getting them right at the outset.
For Asymptote I'm less sure how to do this. The Asymptote code doesn't make it into the LaTeX file. Instead, we run the pretext script to generate pdf images for each figure.
That script inserts all the LaTeX macros, as well as the asymptote-preamble, into each asy file.
I don't know a better way to do this. And changes to the top level file are so infrequent that this was the cheapest option from my point of view.