Response below copied from OP's cross-posted question/thread. Copying here so as to delete original response.
Have you looked at Essential readings about the geodatabase documentation? If not, I strongly encourage you to spend some time reviewing that material. The Esri geodatabase (see What is a geodatabase?) is the native data structure for ArcGIS. The Esri geodatabase conceptual model exists independent of DBMSes, but it can be implemented in enterprise DBMSes to create an enterprise geodatabase, which has additionally functionality over file or personal geodatabases.
Although ArcGIS can connect to DBMSes and consume spatial data without having ArcSDE/SDE/geodatabase middleware, the Esri geodatabase extends that functionality. For example, numerous Esri datasets or models (Raster dataset, Mosaic dataset, Geometric network, Parcel fabric, Topology, etc....) are implemented within the Esri geodatabase. If you want to work with those types of datasets or models, the choice is made for you. Also, there are limits to editing or manipulating spatial data in DBMSes when not connecting to an enterprise geodatabase.
As simple as your question seems, it is too broad or open ended to get succinct answers. If ArcGIS is going to be your primary platform for consuming, manipulating, and managing spatial data; I can't imagine not using Esri geodatabases. If ArcGIS is simply a client for consuming spatial data for analysis, visualization, etc...; then it might work to forgo Esri geodatabases. In the end, what really drives the decision is your requirements, and no one can offer worthwhile advice without knowing more about those requirements.