AppEngine is a "Platform as a Service" although they are re-phrasing it as "serverless" these days. It has benefits like you dont have to worry about the "roll your own" side of things - you just deploy your code and it runs (Google takes care of the servers - patches, security etc).

The "killer" feature of AppEngine is the scalability - i.e. your small-scale virtual or shared servers are ok for small usage, but what happens when you have more traffic than that VPS can handle? Perhaps call the hosting provider and ask for more CPU/Mem/Bandwidth, or maybe move to a new bigger server elsewhere. But that takes time, and what if it was just a spike and now your traffic is back to normal but you've got a beast of a server running the site. You could be wasting money even with the small VPS sitting idle for most of the day if you have very infrequent traffic.

With AppEngine it just scales automatically to handle the load, from zero QPS to many thousands QPS and back down to zero again without any effort from the developer apart from the initial config (... of course you can limit that scaling too if you want to control cost).

There are also a bunch of other nice-to-have features around traffic splitting (i.e. A/B testing), some nice task-handling tools (e.g. cron-style stuff, task queues), centralised logging, and the easy plug-ins to the other Google Cloud stuff.

Deployments themselves are not especially "quick" and you have to do a few things "the appengine way", but the trade-off is you have nothing to manage on the server at all.

I've used AppEngine a lot over the past few years (mainly Java and Go) and its pretty decent I've found, although I do use a lot of the other Google Cloud services (e.g. their database stuff) so it makes sense. App Engine has been pretty rock solid in my experience.