Hi,
Thank you very much for the details, it helped a lot!
This was due to a very rare known issue. It had been reported once before and luckily I managed to find the culprit. Basically when the plugin tried to generate the stylesheets, it looked for taxonomy filters, but for some reason these taxonomies were registered later on, so WordPress reported as if they didn’t exist. This is very likely happening for a difference in plugin activation order. The default code then ignores the filter and moves on, but that leads to some missing styling.
I have made a tiny correction to the plugin code (file: plugins/ajax-search-pro/functions/functions.php) so the code ignores the missing taxonomy and proceeds with the style generation. After saving the search settings the styling is now repaired.
The only difference in styling I see is some custom CSS, which is not present on the dev, but present on the live site.
This change is going to be included in the upcoming patch of course. If you update to the live, please make sure to copy the plugin folder from the dev to live so the changes propagate.