What actually happens when we work out a problem with a coarse mesh in one case and a finer mesh in the other case in aircraft structural analysis?

Mesh type, size, controling influences the result of FEM to a great extent.

Fine Mesh provide more accurate results.

Coarse Mesh provide less accurate result.

In simple model, difference is slight. But in complicated shapes, fine mesh is required. At edges or where difficult to get a proper shape like corners, cutouts, choose manual meshing to get accurate result.

The results of the Finite Element Analysis should not vary much with the how coarse or fine is the mesh. Hence, grid convergence study is recommended to check for the variation in results. For large structures, to reduce computational time and space requirements, it is recommended that fine meshing is done at the critical locations namely stress concentration zones, zone with high strain energies etc. One can alternatively use the grid adaptation method for better result. 

In practice, variation of5% in results is accepted when validating the code with test/experimental results.