I'd also be interested in hearing other ways that the Google Webmaster Tools have helped you with your web site.

One day I looked at WMTs for one of my accounts and found that all the keywords were wrong. I started looking around and realized my site had been hacked, I might not have known it for a dangerously long time otherwise. I make sure I check my sites with WMTs all the time now.
Otherwise it handily points out a lot of errors, like bad redirects, missing pages, sitemap errors, etc., and it shows a lot of things that Analytics doesn't because it takes into account all pages, not just those with the tracking code on them.