True, Awstats may have more features, but I prefer Statcounter as well. Seems to be the easiest to use and the most accurate.
I like Awstats over Google Analytics because you can update it anytime you want so its instant. It also seems a lot more accurate. Although Google Analytics is a bit more detailed.
I am using both Google Analytics and StatCounter. But I am more comfortable and preferably used Statcounter so easy to use and accurate.
Probably I am a little bit paranoid, but I am afraid of giving all the data to Google. GA is not for me. My choice goes to something more private.
Nobody except Google by itself can tell this for sure... But I think that it is .... mmmmm.... not very reasonable to give all your financial data to the company who sell you advertising space (I mean AdWords bids, for example)