Search Console for google.

you give google a signal to do quicker index search on your page, and you can check if there are any defects or things you can make better to optimize your page

its a tool from which you can watch what is happening with your websites.

Google Webmaster Tools (GWT) is one of the main mechanism for the webmaster to communicate with Google. Google Webmaster Tools also helps the webmasters to identify the issues within the site.

Google Webmaster Tools is a great resource for any site owner. It’s designed to inform webmasters about how Google bot is interacting with their website. It gives detailed information on broken links, popular keywords and how often Google is stopping by.

Never used the Bing tool, Google is the best.