Can somebody tell me what the difference between AWstats and Microsoft Analytics is? Which one is better?

The main thing is that MS Analytics is Javascript based, so you only get a "hit" if the browser opens a page containing the javascript and the browser has Javascript enabled (and isn't blocking the Tracker script).

AWStats reads the server logs, and counts every call to the server, from robots, crawlers and other automatic browsers as well as valid users. It also tracks calls to files (like downloads, pdfs, docs, etc.) that do not contain the tracker javascript. For this reason AWstas will always report more calls and visitors to your site... It can also give good insight in the downloads on your site, and calls to images that are loaded on other websites. In which case you might want to check out those sites as they might be stealing bandwidth.

AWstats does have less functions in data-analysis, like paths, funnels, ROI and tracking of campaigns (but I don;t think you are Advertising and marketing if you are asking this question).

Hope this helped!

Use MS ACA or GA if you run campaigns or have a business site. AWstats is good for small personal sites.