One thing about the WooCommerce results page is that it can only display products as results, and nothing else. So in case you have other types of results, like posts, pages, taxonomy terms, tags, categories, users etc.. those will definitely not display there. This is one possible cause.
Another thing I can think of, maybe the theme(or another plugin) has a built-in search override which will always override the results passed from ajax search pro, but it's impossible to tell unless I'm able to see it.
It can be also as simple as something with the plugin configuration, but that I can't tell either without seeing it. If you want, you can leave temporary back-end and FTP details, and I will check on the config, and debug though the code via FTP if needed.Best,
If you like my products, don't forget to rate them on codecanyon :)