BizTalk Server is not a 'bus', it's a 'broker'. The word 'bus' indicates a subsystem that transfers data between components. You might say that BTS does exactly that, but it does it by sitting in the middle of a hub and spoke model, distributing messages to appropriate receivers. Since it often plays a very active role in the distribution of messages, the broker label is more appropriate.
This basically also gives you a pretty good idea about the differences between the Azure Service Bus and BTS. The Azure Service Bus is basically just a data transfer subsystem that enables you to connect various endpoints together across network topology barriers (it enable you to use callback channels to invoke services behind firewalls).
BTS, on the other hand, doesn't do that. Instead, it has these Message Broker features:
- Orchestration
- Message translation
- Content-based routing
- Durable messaging
- Business Analytics Monitoring
- Operations support/diagnostics
- probably some more I forget right now :)
BTS and the Azure Service Bus are really two entirely different things, and might be used complementarily. For example, it's easy to conceive of connecting two BizTalk Servers via the Azure Service Bus.