Microsoft AI is a first party solution Microsoft built, Databricks is based off of Apache Spark that we will manage for you in Azure. One of the things we are trying to do in Azure is meet the customer where they are most comfortable. In a way it's similar to how we have Service Fabric (Microsoft Service) and Azure Kubernetes Service (AKS). Both allow you to run microservices but one is a service we developed and the other is a managed Open Source Project we support.
When to use each is more of a question of preference and skillset