For a web application, should I include a "search" button on a search as you type control, and if so for what reasons?
there are primarily two types of implementations for a dynamic search
if, as the user is typing, the search results are being displayed full page, then there is no need for a 'search' button. a very well executed example of this is the search interface on medium
however, if you are displaying the search results in a drop down menu, providing a 'search' (or 'go') button can be helpful. in this case, the search button can be used to display the results in a comprehensive full page layout with additional information. this is a pretty common interface on social networks. e.g. facebook, twitter, linkedin, etc.
I would say that a search icon would be important, so that the user knows that the text field will perform searching. Having a search button seems redundant though. The button that it is much more useful to include is a clear button. It can be nice to have the search icon where the clear button will be when the text box is empty, and then replace it with the clear button once the user starts typing. Something roughly like this: