Google analytics is more likely to be a realistic figure, as it is only triggered when a page with the tracking code is loaded. Awstats will usually overestimate, as it includes every requested file in the log regardless of whether it was actually a proper visit or whether it was just an image being included on someone else's site.
For example, both Analytics and Reinvigorate both correlate with a pageview count of ~8000, whilst Awstats is over double at 16,111. Similarly, in both your screenshots the 'visits' number is pretty consistent for them both, suggesting that is likely to be a more statistically accurate figure.