As an Azure customer, you enjoy its capability to deploy and manage workloads across many different services and regions. You are responsible for managing those Azure resources, including monitoring for problems. However, the underlying hardware and software supporting Azure (e.g. Azure’s physical hosts and network) are managed and monitored by Microsoft engineers.
Azure Stack runs as an integrated system in your own datacenter so the model is different. When you adopt Azure Stack, you enjoy the same capability to provision and consume workloads, but because Azure Stack services and hardware reside in your datacenter, you are responsible for managing and monitoring the Azure Stack environment to ensure system health and reliability. These tasks are taken care of by a new role in your organization - the Azure Stack operator.