Accuracy can be similar or different, it is not an issue specific of FV or FE methods but depends on the specific implementation (shape function degree, flux reconstruction).
It is about the resolution of the final output, one is looking at and the specific application. For stress, strain and structural performance, it is hereby important to understand long term implications, FEM is required since elements are looked at. When you are looking at problems of Fluid Dynamics or problems of control volume, you use FVM or CFD methods. Although this answer is really generic, you have lot of papers online where a quantitative and qualitative comparison is being made for the same application using FEM and FVM.
It's a good question but not a specific one. The answer depends on the specific solved problem and the used schemes. For FEM, the accuracy of solutions are dependent on the mesh size, the accuracy of scheme. For different schemes, the accuracy can be totally different.
The accuracy of FVM methods are dependent on the flux reconstruction, the mesh size and so on.