The best things is that Google will email you when your site is having issues. This is huge.
How do you know when your website is having issues with the search engine? You don’t. Google needs to tell you.
But there’s also a ton more it does. I’m just going to give you a few highlights.
Search Queries Reports
Since Google Analytics doesn’t give up keywords people are using to search with when they discover your website (only if they are logged in), Google Webmaster tools will provide these search queries regardless if they are logged in or not.
The only issue is that it’s not tied specifically to their visit. 🙁
But you will find a ton of information on your keywords and landing pages like impressions, clicks, click-thru-rates and rankings.
There’s a ton of information to be found here and here’s a great article from LunaMetrics explaining a lot more of it.
You’ll also want to connect your Google Analytics and Google Webmaster Tools. It’s pretty easy to do. Just go to Acquisition -> Search Engine Optimization -> Landing Pages.
The only thing you need to worry about there is that only one account Google Webmaster Tools account can be connected to one Google Analytics account.
Links to your website