If you are looking for a basic website analytics tool that provides some good, high level information about who is visiting your site and where traffic is coming from, Quantcast is a good tool. It's a bit less intimidating than a tool like Google Analytics as there are no custom dashboards.
Free services provide a wonderful resource for people on tight budgets. If willing to put forth work in data managing/analysis and If basic high traffic information or drilling down specific users in low traffic situations is needed, then StatCounter may fit your needs. If automation and low overhead are a high priority then some more expensive packages can provide the same and more information with less hassle. Because StatCounter is free, there is low risk to see if StatCounter works in your specific usage scenario.
For the free version data logging is limited to two weeks, which requires manual pulling of the data (CSV or excel formats) on a regular basis which can become tedious.
For offline graphing/presentations, data must be re synthesized/visualized. This typical requires someone manual manipulation the data or develop scripts or coding to improve the collection of the data for offline presentation and records.
Lacks some of the pretty visuals that can be seen in competitors such as Google Analytics.
Lacks user specific in depth tracking across multiple services (mailchimp, google etc), like more fully featured platforms (Kissmetrics) are able to provide.
I have not had any issues with malfunctions when using statcounter, so that would lead me to continue using the program. I would like to see more social media measuring tools, but again, we don't use the program to dig really deep. We're just looking to figure out who is reading what articles on our website are the most liked. And it helps lead our editorial team in the right direction.
I think both Google Analytics and Quantcast Measure are useful to have at the same time. You can see how some data differs, but both services give different kinds of data that can be useful for research and marketing purposes. The essential features of both services are free, so there's minimal risk in trying out both tools.
Statcounter excels with the simplicity and efficiency of the analytics. You can see exactly what's going on on your sites without the need to dig into complex google analytics data.