What is the practical difference between the fgdb API format and fdgb ArcObjects in FME? When to use each of them in FME?
This FMEpedia article gives a brief comparison: Comparison of FME readers and writers for SDE and Geodatabase.
This FMEpedia article also describes version compatibility for the various geodatabase readers and writers: Notes on FME and Esri Versions and Compatibility
Notably, the File Geodatabase API is only able to work with 10.0 and newer geodatabases.
Use the ArcObjects format when you need to work with pre-10.0 GDBs or use any geodatabase features not supported by the File Geodatabase API (see pages 19-20 of this presentation for an overview).