When I was green at SEO many years ago I fell for the old rank checking software nonsense too, it happens to everyone in the beginning. After a couple of months of comparing those reports with analytics it quickly became obvious that the ranking reports were nothing more than a badge of honor "look at all my top 10s!" and had nothing to do with how well the site was performing. In reality how well you rank in general is meaningless, it's how well your site performs for the keywords sending traffic that means everything.
Analytics tell you pretty much the same thing you used rank checking for because changes of both content and structure will result in increases or losses of traffic for terms and/or you'll see new keyword terms coming into play driving traffic.
Knowing what keywords people are searching for is the most important to know how they're going to find your site and optimize for that.
If you want to know what keywords to focus on you can get a ton of free intel from AdWords, Google trends [google.com], even the search engine suggestions as you type in the search box and they all show you top keywords that get the most traffic.
Once you've optimized for what people search for analytics will show you how well you did and you can literally tweak your SEO from that information alone.
However, if you just want bragging rights, run those ranking reports.
When I want bragging rights, I run my banking reports. ;)