App Service storage is completely different than Azure Storage (blobs/tables/queues).
App Service Storage
For a given tier size (e.g. S1), you get a specific amount of durable storage, shared across all instances of your web app. So, if you get 50GB for a given tier, and you have 5 instances, all 5 instances share that 50GB storage (and all see and use the same directories/files).
All files in your Web App's allocated storage are manipulated via standard file I/O operations.
App Service Storage is durable (meaning there's no single disk to fail, and you won't lose any info stored), until you delete your web app. Then all resources (including the allocated storage, in this example 50GB) are removed.
Azure Storage
Azure Storage, such as blobs, is managed completely independently of web apps. You must access each item in storage (a table, a queue, a blob / container) via REST or a language-specific SDK. A single blob can be as large as 4.75TB, far larger than the largest App Service plan's storage limit.
Unlike App Service / Web App storage, you cannot work with a blob with normal file I/O operations. As I mentioned already, you need to work via API/SDK. If, say, you needed to perform an operation on a blob (e.g. opening/manipulating a zip file), you would typically copy that blob down to working storage in your Web App instance (or VM, etc.), manipulate the file there, then upload the updated file back to blob storage.
Azure Storage is durable (triple-replicated within a region), but has additional options for replication to secondary regions, and even further, allowing for read-only access to the secondary region. Azure Storage also supports additional features such as snapshots, public access to private blobs (through Shared Access Policies & Signatures), and global caching via CDN. Azure Storage will remain in place even if you delete your Web App.
Note: There is also Azure File Storage (backed by Azure Storage), which provides a 5TB file share, and acts similarly to the file share provided by Web Apps. However: You cannot mount an Azure File Storage share with a Web App (though you can access it via API/SDK).