Remote Desktop Not Connecting for Windows Azure Virtual machine (Web Role/ Worker Role)

Check what is the current osFamily version that your Virtual machine using. If the VM is using osFamily="5" then RDP password should follow below rules. 

1. The RDP Password for OS Family 5 must be a minimum of 10 characters.
2. Password should not containing the username

Setup a password as per the above rules and try to RDP the server.

For more Details you can check: https://docs.microsoft.com/en-us/azure/cloud-services/cloud-services-guestos-update-matrix#family-5-releases