Sry, but after reading the accepted answer, I thought about adding another one...
Have you ever, ever, ever, ever, ever, ever thought about privacy? AW stats is all in one place - your own server. It count statistics and can be perfectly private.
If you decide to use google analytics it means, you provide ALL the data about your website visits, ips, tracking, subpages, etc to 3rd party corporation and you are never sure if those statistics are safe. But you can be sure that those statistics will be used against you, and that google will earn money from them in the future. I know google is a monopolist, but you should never serve such an answer, without pointing this outout.
The difference between AW stats and google analytics is like a difference between making your dinner from fresh anonoymous-bought ingredients at home with closed windows and curtains, or buy a chineese 3 minutes fast soup over the internet using your paypal and visa codes in plain text in url. The second one is easier, but this is THE ONLY PLUS.
AWstats are your own stats, while google analytics is "sending your stats to someone, for free, tell someone all about your visits, traffic, clients, phrases, and all". And someone will earn money from that stats and give you some kind of "nice" instead. Is it worth...?