Site Catalyst. Beats the snot out of Google Analytics...but it'll cost you an arm, a leg, and your first child to use it. They may also demand annual blood sacrifices. A client of mine is using it - they're paying $25k/year for their licenses.
You can use this 2 webstats, it will not any harm to your site.... If you use this can check for statcounter and google analytics status of your traffic.
I use both Awstats and GA for monitoring my site. But I still prefer GA because it provides more detailed report.
I can recommend both, BUT Google only if you want to give them all the information about your site's traffic, click through rates, time-on-site, possible conversion etc. They will use this information to set your adwords prices. If you do not use adwords and never intend to, then go for it. I myself use an in-house developed stats and conversion tracking software. If you use Wordpress simply install the Statpress plugin - it rocks.