SQL Server Diff not restoring from Azure BLOB storage
Most likely, you have multiple Differential backups in your DB_NAME_DIFF.bak file, and the restore is picking one that occurred before the most recent FULL.  As I hinted in my comment, and Scott stated in his.

Edit the following to check for multiple files in a backup device:

RESTORE HEADERONLY   
FROM DISK = N'C:\AdventureWorks-FullBackup.bak'   
WITH NOUNLOAD;  
GO