Just use Google. Most external counter sites use javascript. Now here's some interesting facts.

Javascript stops the loading of any other elements on a page eg images, adverts, css etc. until the javascript has being fully downloaded. All Javascript does this. That's why your always encouraged to put the javascript in the footer so the visitor can start looking at and reading the page while the javascript is still downloading.

The Google javascript tracker has parallel download capability. That means it won't stop other things downloading. It's the only javascript tracker you can put in the header of your site and it won't slow it down and therefore gives more accurate readings.

None of the javascript based trackers work if the visitor is using noscript or if their browser doesn't support javascript. In reality this amount is tiny.

Now what can happen is people put the javascript tracker in the footer. This means if the person closes the page before the page is fully loaded then it won't be counted. Awstats will record the vistior as it records people even if the page doesn't get a chance to load which is why their always higher but the external javascripts won't count. If you've the Google code in the header it will record more of these people.

Hope that clears up some stuff as I know if I didn't post this their'd be all sorts of crazy posts saying Google isn't accurate or awstats is the best or some other wild theories.