If you compare the two options FBX is considered to be a higher quality modern 3D exchange format that retains more information while OBJ is a simple model format that is popular for 3D data exchange.
OBJ has it’s origins with Alias Wavefront, a simple data-format that represents 3D geometry — namely, the position of each vertex, the UV position of each texture coordinate vertex, vertex normals, and the faces that make each polygon defined as a list of vertices, and texture vertices. Vertices are stored in a counter-clockwise order by default, making explicit declaration of face normals unnecessary.
FBX® is a proprietary file format originally developed by Kaydara, a 3D asset exchange format compatible with many 3D tools. FBX facilitates higher-fidelity data exchange between Autodesk content creation packages and supports certain third-party and propriety applications. With FBX it’s easier to transfer files, retain more data and work more efficiently.
Here is a chart that will help you decide which format is better for your needs:
|LightMap | DetailMap | NormalMap||Link||No|
|Multiple Channels Mapping||Yes||No|
|Interchange File Format||Yes||Yes|