Aaron Axelsen

The Advanced Search Widget is designed to be a replacement to the defaut search widget. The Advanced Search Widget allows you to search for only a 
specific post type and also select which fields to query (title, content, tags).
In addition, it is also possible to place multiple widgets onto the sidebar to search through different custom post types.

Advanced Search Widget

Allows you to add a widget to search custom post types along with other options

Code for the People Ltd

Scott (@scottsweb)

WP Ajax Search Widget is a replacement for the standard search widget but instead of redirecting you to a new search page the results are returned underneath the search input with an option to click through to the standard results page.
The widget allows you to choose how many results to show.
<a href="http://scott.ee/" title="WordPress designer and developer" rel="nofollow ugc">a plugin by Scott Evans</a>

WP Ajax Search Widget

Displays instant search results directly beneath a search widget.

Advanced Search Widget vs WP Ajax Search Widget