I can't really tell you whether you should use App Engine without knowing anything about what you need your web app to do, but I will tell you what App Engine can and can't (or won't) do.
App Engine is fantastically good at scaling. It is, in fact, designed to scale web apps to ridiculous lengths first and foremost, with ease of use and number of features being secondary goals.
That's not to say that App Engine doesn't have features, or isn't easy to use, just that if there ever becomes a choice between adding a feature and staying scalable, the App Engine team will choose scalability.
For example, App Engine doesn't have some of the features of a relational database, because those features don't scale to the size of an app that App Engine is designed to support. App Engine doesn't support requests taking more than 30 seconds, because App Engine is designed to serve a web app, not process long-running requests.
In general, when App Engine doesn't support something, it's not because it's impossible -- nothing is impossible -- but rather because it would detract from the scalability of App Engine.
There are workarounds that can be (and have been) implemented to get around this, particularly with things like the task queue, and App Engine is constantly getting new features and new frameworks built on top of it.
App Engine for Business adds SLAs and different pricing, but is otherwise pretty much the same App Engine.