One of the compelling reasons I have come across for using Google App Engine is its integration with Google Apps for your domain. Essentially it allows you to create custom, managed web applications that are restricted to the (controlled) logins of your domain.
Most of my experience with this code was building a simple time/task tracking application. The template engine was simple and yet made a multi-page application very approachable. The login/user awareness api is similarly useful. I was able to make a public page/private page paradigm without too much issue. (a user would log in to see the private pages. An anonymous user was only shown the public page.)
I was just getting into the datastore portion of the project when I got pulled away for "real work".
I was able to accomplish a lot (it still is not done yet) in a very little amount of time. Since I had never used Python before, this was particularly pleasant (both because it was a new language for me, and also because the development was still fast despite the new language). I ran into very little that led me to believe that I wouldn't be able to accomplish my task. Instead I have a fairly positive impression of the functionality and features.
That is my experience with it. Perhaps it doesn't represent more than an unfinished toy project, but it does represent an informed trial of the platform, and I hope that helps.