Since you have mentioned "enterprise", I am assuming that you are referring to Alfresco Process Service (formerly known as Alfresco Activiti Enterprise). The REST API root of the enterprise product is "activiti-app/api/".
Since the enterprise product is built on top of the core engine which is open source, all the following core "Process Engine APIs" are available directly under activiti-app/api/. Process Engine REST API | Alfresco Documentation
There is also a set of Enterprise APIs to support some of the enterprise only features which are available under "activiti-app/api/enterprise". They are known as "Process Services APIs". They are documented at http://docs.alfresco.com/process-services1.6/topics/process_services_api.html
If you have an enterprise implementation, the usage of these APIs depends on your requirement.
- For example: Implementation of identity is different in enterprise and hence the users and groups are stored differently. For the same reason, "activiti-app/api/identity" may not return the results you need. In this case, you will have to use an API under /enterprise.
- Another example: if you want to get a list of executions, you will want to use the core engine API "runtime/executions".
I also noticed that you have "/app" in your question. Please note that those APIs are there mainly for the OOTB activiti-app UI use and is secured using the cookie approach based on the logged-in user, device etc. They are classified as "Internal" APIs and are subject to change in newer versions.
Hope I answered your question!