Azure SQL Database is native to the Azure and offered as Platform as a service (PaaS). The objective of this offering is to reduce the overall costs to the minimum for provisioning and managing many databases. This offering comes with built-in high availability, disaster recovery, and upgrades for the database. Databases run on latest version of SQL Server Enterprise Edition. Many databases can be managed by single IT resource which reduces the overall cost of administrations.
SQL Server running on Azure VMs is categorized as Infrastructure as a service (IaaS). This setup is best for migrating existing applications to Azure or extending existing on-premises applications to the cloud in deployments.
You can use preinstalled SQL Server on VMs or install your own licensed version of SQL Server. This setup is good when you want to run existing applications that require fast migration to the cloud with minimal changes. You have the full administrative rights over a dedicated SQL Server instance and a cloud-based VM.