It sounds like that you might have a Network Security Group(NSG) associated with your VM which is blocking the inbound traffic. By default, Azure blocks all incoming internet traffic to your VMs. If you haven't done it already, you'll need to create an NSG rule to allow traffic on the ports you're using. For RDP, you should allow TCP on port 3389. You can Diagnose a virtual machine network traffic filter problem to walk through and verify what NSG is associated to your VM which might be blocking the traffic.
Please check and see if that resolves your issue. If you are still having issues let me know and we can work with you directly to continue troubleshooting.
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