Despite both programs appearing to give the same information, the user statistics and viewer data can vary wildly between Google Analytics and AW Stats. This is because both programs gather their information in a different way, and for different purposes.
AW Stats
AW Stats gets their information from the raw backend of your website and is the most accurate for finding the number of visitors to your website. If you have posted a new blog post or added a new product to your website, and you want to know how many times that link or page has been clicked on, AW Stats is going to be the best way to find out.
Google Analytics
GA gets their information from Javascript running on the website and cookies that are downloaded by each user visiting your website. Because GA relies on Javascript and cookies for viewer information, it will usually show fewer users than AW Stats, because many users have Javascript or cookies disabled when they browse the web. Google Analytics is most useful for finding out what kind of people use your website. While AW Stats can show you the number of people who have visited the website, Google Analytics can give you more insight as to the average age, gender, and location of your visitors, as well as how these users have interacted with your website (ie, how long they spent on each page). This information can be very useful for finding or targeting demographics for your business.
To make things simple, think of AW Stats like a census, whereas Google Analytics is more like a focus group. Just as the names imply, AW Stats is for finding your viewer statistics, and Google Analytics is for analyzing your viewers.