Q. Are Azure Container Instances the same as Azure Container Service?
A. No. Azure Container Service is an engine to deploy a container orchestration solution to IaaS VMs, for example a deployment of Kubernetes or DC/OS or Swarm. This orchestration environment then manages container hosts which run the container instances. Once Azure Container Services deploy the orchestration solution you are then responsible for the ongoing management of the VMs and the deployed orchestrator.
Azure Container Instances are really containers as a service. You request a container instance to be created based on an image and the container is created for you. You don't see an orchestrator, you don't see a VM, you don't see anything other than your container instance. Azure Container Instances are also billed per second which is geared towards the micro-service nature of containers than may be created and deleted in seconds. When creating a container instance you can select if it should be provisioned with a public IP address.