google webmaster tools helps to monitor your website, it will give you the over view about search queries, backlinks list, also helps to recover your website from google updates via disavow links tool

Google Webmaster tools is very importante for that Google index your content as quickly as possible

I think it's pretty important. You can get lots of great statistical data from Webmaster Tools and you can also get informed if there is something wrong with your site that could effect its ranking (for example, if you're site isn't mobile friendly)... For as quick and easy as it is to put on your site - You should be using it all the time!