The cloud services provides a range of options from fully managed to less managed services. Less managed services gives more control to the developers. Below are the differences

(Disclaimer: I work in the Google Cloud Platform team, but this is a personal answer.)

All of these are solutions which allow you to host your applications in the cloud. You can view them as a sort of spectrum of control/automatic management. There's also another thing to consider: App Engine comes in two environments, "Standard" and "Flexible".

So the spectrum ends up being:

• App Engine Standard: limited runtimes (Python, Java, PHP, Go), very simple to get up and running, auto-scaled etc. Focused APIs designed specifically for App Engine.
• App Engine Flexible: anything you can put in a container, auto-scaled etc.
• Container Engine: you design your application in terms of micro-services, specify how they're scaled etc, but let Container Engine (the Google Cloud Platform implementation of Kubernetes) perform the scaling
• Compute Engine: Hosted VMs with benefits, basically. Some features such as live migration, auto scaling within Managed Instance Groups, but it's much more "bare metal" than the above.

Basically, you need to decide how "hands-on" you want to be. The more control you decide to take, the more work you have to do but the more precise tuning (and flexibility) you get.

Additionally, App Engine Standard supports "scale to zero" - if your app isn't getting any traffic, it can be completely dormant. That makes it a great environment for hobby projects.

Compute Engine is more on Infrastructure side. App Engine is more on Application side. Container Engine is more on a cluster management and container orchestration system developed to run and manage Docker containers.