AWS employee here. You would deploy to Fargate because you don't want to have to manage the underlying EC2 instances. You can use Fargate with both ECS and EKS (in 2018).

ECS and EKS are just two different schedulers for orchestrating the containerized services that you want to run. Fargate is the engine behind them that executes the containers for you without you needing to worry about servers.

ECS as a scheduler will always integrate much better with other AWS services. EKS will give you the advantage of being able to run the same scheduler on an on premise or on another cloud.