I will definitely second Google Analytics. AFAIK Development behind the others is slow or non existent. Figures you see in Webalizer or AWStats will be inflated, if what you really want to know is real, human traffic. This is why websites buyers will always want to see Google Analytics stats - if the main value of the site is its traffic.
However, if you find a lot of content that is hot linked for example, analytics just won't tally with log file reporting at all.
Personally, I'm a fan of log files over JS based logging. It might not be able to do quite as much testing of the client browser, but it is accurate.
I prefer AWStats to Webalizer, and I've never used Modlogan. I also download the raw access files.
I don't use Google Analytics.
Also another important feature - a/b testing. Critical for large marketing campaigns, and handy in conjunction with google website optimizer for general usability/conversion testing.
I used to use Google analytics exclusively until a friend told me about a service called Getclicky... lets just say I rarely use Google analytics anymore. Thats how good it is.
TheRaptor - Joe
I'm using Google Analytics for my sites but I have another tool which helps me a lot and I wont give up soon to using it. My helping hand is the Traffic Counter of RankVision.com which gives me info on the visitors I get, which country they come from, where they are being referred from or what pages they visit.
I'm using AwStats and sometime i compare results with Google Analytics, even they differ. I don't know Modlogan, while Webalizer it seems to me a nice version of AwStats.
WordPress Plugins ( just one by now ) and some other stuff