Scalability, Azure Security and Availability are few of the major benefits of Azure DB for MySQL. These can be attained to some extend on premises as well but the service being a PaaS offering, All the administration is fully managed by service itself.
This results in lesser administration and lesser points of failure. And with robust SLA's of Azure Cloud for Availability and Back up, your Data is secure. Also Data is secure at rest as well as in motion.
Here is a document to look at for getting started and get the brief Overview.
Hope this helps.