Your short answer is "Windows Service Bus 1.1".
https://msdn.microsoft.com/en-us/library/dn282152.aspx
Ideally, you're going to write the same code and can deploy to either.
That does not mean there are no differences. Namely "security". Using " Shared Access Secrets" will be safe in both worlds.
Below are 2 URL's to help.
Azure Queues and Service Bus queues - compared and contrasted
https://azure.microsoft.com/en-us/documentation/articles/service-bus-azure-and-service-bus-queues-compared-contrasted/
Service Bus for Windows Server 1.1 Overview
https://msdn.microsoft.com/library/azure/dn282142.aspx
In both Microsoft Azure and Windows Server, Service Bus requires
access tokens for authorizing access to its messaging entities. Both
share the Shared Access Secrets (SAS) authentication scheme for
Service Bus namespaces as well as entities (queues and topics).
However, in Windows Azure, Service Bus also supports the Microsoft
Azure Active Directory Access Control (also known as Access Control
Service or ACS), which is not available on Windows Server.
However, on
Windows Server, Service Bus supports Windows integrated authentication
(domain joined users and Active Directory user groups), which are not
available in Azure.