I'm puzzled that many discussions of Firebase (including the question and answer above) fail to mention what, to me, is a very important difference: price.
Here is the Firebase price schedule.
Here are the Datastore and GAE pricing.
It can be tricky to compare these, but my interpretation is that Firebase is very expensive.
And this should come as no surprise. GAE and datastore have to compete with similar services from Amazon, Microsoft etc., and the competition is fierce. Yes, these services are not as generic as infrastructure and SQL, of course, but they seem to be close enough that the prices remain competitive.
Firebase, on the other hand, is a premium service that competes with other backend services like Parse, and once you decide to use it I think it would be very difficult to switch. It should come as no surprise that Google is pushing Firebase so hard - they are probably going to make a ton of money off of it since they can price it at such a premium.
In my opinion, the upshot of this is that Firebase is a good choice for low volume and high-margin services, but if you plan to create a typical, consumer oriented, ad supported service that would depend on large volume to make money, then the cost of Firebase may kill your profit.
2017-10 Addition:
I looked at Firebase again with the recent release of Firestore.
I think it is important to be aware of another issue: using Firestore for an Android app means using the Firebase client library which is heavily dependent on Google Play Services, which means you can't deploy to non-Google devices including Amazon Fire tablets and (I believe) the entire Chinese market.