We are currently considering acquiring a Web Analytics solution for our on-premise SharePoint infrastructure consisting of pretty much all the latest versions of the platform: 2010, 2013 and 2016. The reason is beside the point but suffice it to say that we do not consider the SharePoint analytics features to be anywhere near enough compared to what the latest solutions on the market have to offer.
After having carefully reviewed tons of various systems (many of which, such as Google Analytics, do not fit the on-premise model), we found that only a handful of them claim to provide SharePoint-specific features/reports. Most of these systems dismiss the issue on the grounds that SharePoint sites are, after all, well... web sites and that there's basically nothing special out there about them, except maybe developing a few additional reports.
There are however other systems that take the SharePoint market segment pretty seriously and offer dedicated solutions for "integrating" their web analytics solutions with SharePoint. One such example is Piwik with their support for the Minimal Download Strategy...
So, here is the question: how justified is this integration argument? Is it worth the money or any other decent enough solution out there could be easily adapted to SharePoint by just specifying a few additional variables and visualizing them in custom reports?
First of all it's about privacy and security. If you do not care, then you could use Google solution.
On premises solutions offer you a "closed solution" where you do not need to send data outside of your organization.
Second, it's about support for SharePoint structure.
See, SharePoint has its own Database and it has a dedicated DB and tables that contain the statistical data designed to track SharePoint features.
These products know how to integrate, read this data and display it in a meaningful and user friendly format.
Others do not.
Also, there is a difference between products that claim to have SharePoint integration.
See, the logs contain entries that are not necessarily in sequence. The product that can help you is the one that can recognize a context and not just take the log entries as separate entities and "flag" or categorize them. You need to make sure that the data you get from SharePoint logs is meaningful and helps you see the business process.
I hope this helped clarify the issue a bit.
P.S. I will not name or recommend ANY product. This site should not be used for commercial promotion of any kind.