The Google News homepage uses a special search box with a few interesting features. Google only shows a list of suggestions and doesn't use Google Instant, even if this feature is available for the Google News results pages. The last suggestion lets you search the web and you can use the up arrow key to select it or just press Shift+Enter. A similar feature is available in Gmail, but you can't use the Shift+Enter shortcut.
When Google redesigned Google News and other services, the "Web Search" button seemed extraneous and it was removed.