Not sure how I missed this when it originally aired, but this Azure Friday video from last June does a great job of showing off one of the benefits of Azure Web Sites, namely free analytics.
They come in two flavors, server-side stats, which are enabled as soon as you create your site, and client-side stats, which require that you add a small snippet of JavaScript to your main site layout.
Once the feature is enabled, you can get rich information about visitors to your site, including browser stats, page response time (and you can set up alerts to make sure you’re aware when stats exceed certain thresholds), and more.
As someone who used to use a separate analytics package (Piwik) to capture this data, it’s much nicer to be able to have it all in the same management portal (note that the advanced analytics features are only available in the newer “Preview” portal at the time this post was written).