In general, these APIs just work on different levels of abstraction, and using a high-level APIs is often recommended, as sometimes the layout on the repository level is well-known, but technically an implementation detail and could change with every release. So, start with the API on a WCM level, and all resource-handling should be done using the Sling Resource API.
The only reason to use the JCR API is only the use of nodetypes, which is often required, but not supported by Sling.
As Praveen stated - the API you use all depends upon what you want to do. AEM is a large product with many pieces of functionality. Hence is the reason why there are many APIs for that reason.
For example - if you wan to programmatically work with DAM Assets - you use the AssetManager API. If you want to create a custom workflow - you use the AEM Workflow APIs. If you want to query the JCR - you can use QUeryBuilder API or JCR API. If you want to work with databases, you can use the DataSourcePool API.
We have a lot of community articles that get you up and running with the various AEM APIs. You can see them here:
My only question is if the same specific function could be achieved by JCR API, Sling API, WCM API and Jackrabbit API, which API should the developer choose?
In general, these APIs just work on different levels of abstraction, and using a high-level APIs is often recommended, as sometimes the layout on the repository level is well-known, but technically an implementation detail and could change with every release. So, start with the API on a WCM level, and all resource-handling should be done using the Sling Resource API.
The only reason to use the JCR API is only the use of nodetypes, which is often required, but not supported by Sling.
