You cannot use quotes in your search withing the module search. Quotes in the search term will work in the components search. I have contacted the owner about this, but have not heard back.
The other issue is more of a nit-pick. The search results only display 4 results. In my opinion you should be able to enter how many results you would like displayed.
All in all, this is better than the stock Joomla Search.
Thank you for the review.
I replaced the old AJAX search API with the new Google Custom Search API.
I fixed the bug with the quotes.
Now you are able to list more than four results.