The ability to personalize content for each of your online visitors is part of creating an engaging customer experience. You cannot accomplish this with Google Analytics. Google's policies forbid the aggregation of personally identifiable data, such as information entered into an online form.
Sitecore DMS is a server-side operation. If you want to know how many times a particular user viewed a specific page during a session, Sitecore analytics can tell you. Google Analytics cannot; it does not offer the "deep drilling" at the user level. Running on the server side also improves the accuracy of the data although the difference may be insignificant for your purposes. Most notably, users can "turn off" tracking in Google, thereby remaining invisible to GA. As for timeliness, since GA is not a server-side operation, it will always lag a little behind on data as well.
The basic Google Analytics package is free to everyone. Sitecore is not. Google does offer an upgraded version that adds more functionality, but for the price, Sitecore is often the better return on investment.
If you need the ability to create triggers to generate automated email campaigns, you will not find the option in GA. Sitecore, however, has a built-in module designed for the purpose.
Both Sitecore and Google Analytics do an excellent job of aggregating bulk data. Sitecore allows you to assign values to events, triggers or pages to determine which have the most value per visit. Options for assigning values in GA are more limited and must be coded for every value tracked.
The reporting tools in Google Analytics are easy to use and feature an intuitive interface that requires little time for even novices to master. However, the executive dashboard in Sitecore offers a greater variety of reporting options and filters.
The first thing to remember is that you can do little more than make educated guesses if you have no analytics at all. Therefore, if you need to start acquiring information quickly -- i.e., you have no other tool in place -- use Google Analytics. It is fast and easy to implement, and you can start receiving at least some of the information you need in a relatively short time. This also gives you a set of "training wheels" to begin an evaluation of exactly what you need in the way of analytics. You might find that GA offers you everything you need at this point, or it can help you identify features that you want to be sure you can have in Sitecore. Since Google Analytics is free, you do not need to worry whether your results will justify the expense -- if you gain any benefit at all, you have a good return on your investment. Therefore, if you rely primarily on clickstream data collection and are operating on an extremely tight budget, you want GA.
If you are planning a Sitecore implementation or already have Sitecore, using DMS is only logical. It is designed to integrate with all other Sitecore features to maximize leverage of data. The marketing center and executive dashboard are powerful features, and Sitecore engagement analytics give you access to the "deep" data you need to offer customers a truly personalized experience in real-time.