Using Google Analytics has for example these disadvantages:

• It's not possible to customize it or implement new reports - only the reports designed and implemented by Google are available.
• It does not have access to internal parameters of your application, so it is not possible to log them. For example I need to store user details (user ID, etc.) or request parameters (language, important HTTP headers, etc.) so that I can use them when building reports.
• The collected data are not under your control - Google owns them effectively, and there are people who believe Google is pure evil (No, I don't think so, but I'm not sure Google is 100% trustworthy eiher. A well known quotation says "Just because you’re paranoid, doesn’t mean they’re not out to get you.").
• And another security / reliability reason - the logging is performed by a Javascript library, included from a different server. If someone is able to fob off his own version of the library (a good old fashioned "man-in-the-middle" attack, DNS hijacking, etc.) he may do a lot of nasty things to visitors of your website. And even if there is no attacker, the google server may not be reachable for some reason - this will cause serious problems to your site (OK, you may create a local copy of the library, but who does that?).

This does not mean I don't like Google Analytics - I just don't think this is suitable for all projects.