Finally, found the issue and resolved it.
The Azure Webrole is not installing the Agent if it is already installed in the VM or in Cloud. So once you deploy the Cloud package for the first time, need to make sure that the Agent installer is replaced with the one downloaded from AppDynamics controller.
So there is an inconsistency between the versions of controller hosted in AppDynamics Cloud and the one in AppDynamics Cloud Nuget packages.
I've logged into the Remote Desktop of WebRole and verified the version. Found that the Agent in controller is latest, and manually uninstalled.
Re-Installed the older version and it worked.
This ticket can be closed.