Personally, I don't WANT to give Goog deep metric data about all my sites. So I use Analytics on only a few. StatCounter doesn't keep a long history - but what it tells me in REAL-TIME and in the Zen of 'Be Here Now' - is alot more informative than Analytics long history of things like where traffic USED to come from (but doesn't anymore), keywords that USED to give you traffic (but don't anymore)... Besides I like to keep Goog guessing. They already control too much.
I use both of them, i like statcounter because it's real time and google analytics for detailed analytics
Stick to one preferable Analytics. It gives you more in depth view of your sites traffic. Demographic data is the best part of it. There is also technical details on browsers/os/connection speeds. These will help you design and redesign your site to fit more to these technical s. eg. If you site is viewed more from a mobile device, you can create a separate mobile version of your site. Always keep you faithful visitors happy.
but the difference is that in stat counter you can check about the links on ur adsense ads and can checkout things related to the activity on your website... but with google you can check from which browser people are visiting your website or from which keyword they are getting to your website and many other options are available with google...