In short: if you ...
- have an application that needs to store or access files in the cloud, use Blob Storage
- need a file share that can be used by, for instance, a server, use File Shares
Azure Files shares can be mounted concurrently by cloud or on-premises deployments of Windows, Linux, and macOS. Azure Files shares can also be cached on Windows Servers with Azure File Sync for fast access near where the data is being used.
This means a File Share is, somewhat simplified, similar to a network share you would have in a local environment.
Azure Blob Storage helps you create data lakes for your analytics needs, and provides storage to build powerful cloud-native and mobile apps. Optimize costs with tiered storage for your long-term data, and flexibly scale up for high-performance computing and machine learning workloads.
This means Blob Storage is what you need when you're building powerful cloud-native and mobile apps.