I've tested Google analystics to see if it updates when I or my business partner visit from various locations. (work, internet cafe etc) In short it misses an awful lot of visits. At the ICafe they had Kaspersky AV installed which blocks the functions required to have that visit listed in Google stats. It seems likely that both Norton and Macafee do the same. Finally, Firefox - as of an update in mid 2009 will not allow the data to be collected by Google Analystics at all - Now, even IE in certain circumstances will prevent GA from collecting the data (if people choose the "private" session).
Of the eleven locations we tried to log from, GA only collected three. The other eight were never recorded. We know they happened, and were from unique (and pretty distant DNS's) because we did them.
The raw data collection on your server does not require any special conditions to work.
So the visitors tab in AWstats will give a more accurate visitors figure, but if you want to know what the few visitors who do not have any of the above actually DID while they were on your website, then GA is best.
My advice then, use the web host data colection for raw numbers (and make sure you know what each stat means, Visitors and Pages are best, with pages being a decent multiple of Visitors - depending on how big your site is and what you are trying to achieve)
To see good stats on what those that allowed it did. i.e bounce rate, location etc - then Google Analytics.
The problem as I see it is that there are so many conditions as of July 2009 that stop GA from recording anything at all, that it may soon become pretty useless altogether. I can't see Gooogle putting up with that "Turn your virus checkers off, we want to record you" lol