App Engine is a Platform-as-a-Service. It means that you simply deploy your code, and the platform does everything else for you. For example, if your app becomes very successful, App Engine will automatically create more instances to handle the increased volume.
Read more about App Engine
Compute Engine is an Infrastructure-as-a-Service. You have to create and configure your own virtual machine instances. It gives you more flexibility and generally costs much less than App Engine. The drawback is that you have to manage your app and virtual machines yourself.
Read more about Compute Engine
You can mix both App Engine and Compute Engine, if necessary. They both work well with the other parts of the Google Cloud Platform.
EDIT (May 2016):
One more important distinction: projects running on App Engine can scale down to zero instances if no requests are coming in. This is extremely useful at the development stage as you can go for weeks without going over the generous free quota of instance-hours. Flexible runtime (i.e. "managed VMs") require at least one instance to run constantly.
EDIT (April 2017):
Cloud Functions (currently in beta) is the next level up from App Engine in terms of abstraction - no instances! It allows developers to deploy bite-size pieces of code that execute in response to different events, which may include HTTP requests, changes in Cloud Storage, etc.
The biggest difference with App Engine is that functions are priced per 100 milliseconds, while App Engine's instances shut down only after 15 minutes of inactivity. Another advantage is that Cloud Functions execute immediately, while a call to App Engine may require a new instance - and cold-starting a new instance may take a few seconds or longer (depending on runtime and your code).
This makes Cloud Functions ideal for (a) rare calls - no need to keep an instance live just in case something happens, (b) rapidly changing loads where instances are often spinning and shutting down, and possibly more use cases.
Read more about Cloud Functions
It is not possible at this time to use docker-compose to have multiple application containers within a single App Engine instance. This does seem however to be by design.
Scaling application components independently
If you would like to have multiple application containers, you would need to deploy them as separate App Engine services. There would still only be a single application container per service instance but there could be multiple instances of each service. This would grant you the flexibility you seek of scaling each application component independently. In addition, if the application in a container were to hang, it could not affect other services as they would reside in different VMs.
An added benefit of deploying each component as a separate service is that one need not use the flexible environment for every service. For some very small tasks such as API backends or serving relatively slow-changing web content, the standard environment may suffice and may be less expensive at low resource levels.
Communication between components
Since one of your comments mentions getting instance IPs, I thought you might find inter-service communication useful. I'm not certain for what reason you wish to use VM instance IPs but a typical use case might be to communicate between instances or services. To do this without instance IPs, your best bet is to issue HTTP request from one service to another simply using the appropriate url. If you have a service called web and one called api, the web service can issue a request to api.mycustomdomain.com where your application is hosted and the api service will receive a request with the X-Appengine-Inbound-Appid header specified with your project ID. This can serve as a way a identifying the request as coming from your own application.
Multicontainer application using Docker
You mention many examples of applications including NGinx, PHP-FPM, RabbitMQ, etc.. With App Engine using custom runtimes, you can deploy any container to handle traffic as long as it responds to requests from port 8080. Keep in mind that the primary purpose of the application is to serve responses. The instances should be designed to start up and shut down quickly to be horizontally scalable. They should not be used to store any application data. That should remain outside of App Engine using tools like Cloud SQL, Cloud Datastore, BigQuery or your own Redis instance running on Compute Engine.
I hope this clarifies a few things and answers your questions.
Best Answer
They are different things. App Engine Flexible is focused on application development - i.e. you have an application and you want to be deployed and managed by Google). On the other hand, Kubernetes is more about having your own infrastructure. Obviously, you can also deploy applications in Kubernetes but, as it's your "own" infrastructure, you are the one to directly manage how both the and the application will behave (create services, create scalability policies, RBAC, security policies...).
In this sense, Kubernetes is more flexible in what you can achieve. However, as a developer, you may not be interested in the infrastructure at all, only that your application works and scales. For this kind of profile, App Engine Flexible is more suitable.
If, on the other side, want to manage a complete Container infrastructure (more SRE profile), then Kubernetes is for you.