Ok, I sorted this out.
This probably isn’t particularly likely to help anyone else but, just in case:
This was caused by a typo in the IP for the hostname’s entry in the /etc/hosts file. Once corrected, MaxGalleria started working properly.
A little more info here:
I can see the request and response using Firebug.
On the working server, add_media_library_images_to_gallery sends a request to admin-ajax.php, which responds 264.
On the non-working server, a similar request gets a response of 0.
I am assuming that the 0 is a failure code.
Cursory research suggests that it means either that the action is not set (in the ajax data) or that the action’s callback function cannot be found (i.e., an issue in MaxGalleria when making the call). I’m not sure why this would be true on one server but, not another.
I don’t know if admin-ajax.php does or can do any logging. I don’t really have the time or energy on this project to dive into the WP developer docs, sort out how MaxGalleria works or, hack logging into it to see what is going wrong in more detail.