The Webalizer is very good at mining your Apache logs for stats -- who visits, from what IP addresses, a breakdown of the kinds of domains those IPs reverse-resolve to (.com, .net, .co.uk, .ru, etc.), what their browsers claim to be (which is usually, though not always, what they actually are), what operating system the user is using (again, provided the browser is reporting it honestly), etc., etc. It's completely non-intrusive; nothing special on your web pages, no additional requests to external servers on every page load, etc.
Google Analytics, on the other hand, works at the client level. Every page on your site calls out to Google Analytics via a script tag so that they can capture this information for you (and a great deal more besides). Dead easy to use, but it does incur that extra request to their servers on every page load. Also, note that some advertising blockers will also block Google Analytics; and GA requires Javascript, which some (though not many) may have turned off.
Those would be two of the big ones, each with its different approach. The Webalizer being entirely local (and therefore private) and non-intrusive may be appealing; the scope and analysis features of GA may be appealing.