Is your organization is planning to move your integration solution to Microsoft based platform and you are confused with the option available? Is it really worth to go on cloud? What are limitations and possibilities? What could be challenges? So let’s try to simplify it. It is well known fact that, for each requirement now a day’s more than one option available, maybe I can say lot of options available, and it is quite tricky to figure out best one is best for your requirement.
For all this kind of situations I always suggest to list down your current requirement and your future vision for your platform. Till the time you are listing your requirement let me brief you about two BizTalk based options provided by Microsoft.
BizTalk server is Microsoft’s central platform for enterprise application integration. After first release on 19 December 2000, it has been evolved lot in last 15 years. Now I must say it is most popular integration platform used by lots of big organization across the world. BizTalk server family has 8 versions released till now. Latest on this date is BizTalk server 2013 R2.
BizTalk azure service is Microsoft’s azure based integration platform. It’s quite new and I can say this is still not matured as compare to other cloud based integration platform like Mule soft. But it is there.  
To understand the difference between these two platforms, let’s try explain from point of different stake holders.
Developer-1. You don’t need to install BizTalk server on your local machine, since BizTalk azure service come with SDK which will be used to develop the integration. 2. You cannot use VS2013 since latest SDK support only VS 2012. 3. No orchestrations available with BizTalk azure service 4. Map is bit different with different extension available in BizTalk azure service. 5. Very limited adapters available with BizTalk azure service 6. Deployment done directly on Microsoft azure cloud. 7. Tracking can be done on Azure portal 8. No BAM available with BizTalk azure service. 9. No message box available for BizTalk service.
License- 1. For on premise BizTalk server you need to buy license, but for BizTalk azure service obvious you can pay rent as per your usage. 2. Need Microsoft azure Portal access for BizTalk azure service.
If above is still not clear then try to understand from different technical point of view
BizTalk dominating features

BizTalk server
  BizTalk service
Orchestration
Yes
NO
BAM
Yes
NO
Business Rules
Yes
NO
UDDI
Yes
NO
RFID
Yes
NO

Connectivity/Adapters difference
Obvious you cannot use FILE adapter on BizTalk service, but the common adapters used are FTP, SFTP, SOAP web service and HTTP. Rest other adapters are not available. You can write custom adapters through outbound WCF service.
You can check lots of other difference on different available sites. A good overview can be found on MSDN.
So now you have some info about basic difference between these two platforms. Here is my listings for solution suggested for different scenario
Scenario
BizTalk server 2013
BizTalk service
BAM is needed
Yes
NO
Business Rule needed
Yes
NO
Vendors using on premises applications behind firewall
Yes
NO
Advance tracking needed
Yes
NO
Need to follow Service Oriented Architecture
Yes
No
Already have integration platform and need to upgrade to latest
Yes
NO
Only EDI based integrations
Yes
Yes
Integration platform need to integrate cloud based applications
NO
Yes

Conclusion: BizTalk service is still not matured and lacking lots of needed functionality, which Microsoft may add in future versions. I must say currently lot of BizTalk related stuff is not straight forward to implement in BizTalk service, but still it is possible to implement with some complexity. If you have decided to go with BizTalk azure service for your solution, you can do that but you need to be mentally prepared to write lots of code.