When compared to Google Analytics the primary advantages are the real-time reporting and unsampled data. Google has made vast strides in this area, but Clicky remains a strong competitor as a stand alone analytics solution for your website.
It is incredibly easy to implement, even compared to pasting in a Google script. It is always worthwhile to check your Analytics data against another service to see if there is any discrepancy or other errors.
If you are running an agency and want to white label their service you can do that with Clicky and provide your own custom Analytics dashboards for your clients.
All users are Clicky affiliates, so you can earn enough referrals you can use the paid features :-) not too shabby.
There are a lot of features that are paid that you are probably used to seeing in Google Analytics. Doesn't really show search keywords for impressions like Google Webmaster Tools.
A lot of the features are only available on the paid subscription.
It doesn't really do anything other than fill in the gaps that Google Analytics leaves behind. Real-time was the best thing but they both do that equally well now.