I'm new to these both technologies but have trouble understanding what exactly do these two do different, a use case example will be very helpful.
AWS ECS is the container orchestration service that allows deployment, scale of the containers. Let's say you have 10 apps to be deployed on EC2 machines. ECS will provide you an easy way to deploy and manage them, scale the app when needed etc.
Now, these 10 apps might want to talk to each other. One way is to use the ip address and make an RPC call to the other application. However, this process doesn't scale. What if the machine is restarted or the app or the app is moved to another EC2 machine etc.
So, you require a middleware that manages the mapping of app to the EC2 machine so that the application doesn't need to bother about how to call the other application.
AWS AppMesh provides exactly that middleware. It provides an application level networking so that your service can communicate with other services.