Google Webmaster Tools (GWT) is the primary mechanism for Google to communicate with webmasters. Google Webmaster Tools helps you to identify issues with your site and can even let you know if it has been infected with malware (not something you ever want to see, but if you haven’t spotted it yourself, or had one of your users tweet at you to let you know, it’s invaluable).
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Google Search Console (previously Google Webmaster Tools) is a no-charge web service by Google for webmasters. It allows webmasters to check indexing status and optimize visibility of their websites. As of May 20, 2015, Google rebranded Google Webmaster Tools as Google Search Console.