fdtd vs fem
A good rule of thumb is to have the mesh edge length to be less than a tenth of a wavelength. (This is true in time domain methods as well.) This parameter is called the linear meshing density, and if hold this parameter is held constant, we can compare the three methods. Since everybody likes acronyms, let's call this the LMD.
PDE methods, like FEM and FDTD, use volume meshing. So the number of unknowns increase with the cube of the linear meshing density. For these methods memory and solution time scale proportionally with number of unknowns. Thus memory and time are O(LMD^3). The constant of proportionality depends greatly on implementation and the type of problem. For an antenna problem, especially near resonance, FEM tends to have the advantage.
Integral equations (e.g. MoM) use a surface mesh, and the number of unknowns increase with the square of the linear meshing density. PDE methods produce sparse matrices while IE methods produce dense matrices. For a dense matrix using an iterative solver, the memory and solution time scale with square of the unknowns, or O(LMD^4). However, there are fast methods for integral equations which reduce memory and time to O(N log N) with respect to the number of unknowns. Or in terms linear meshing density, O(LMD^2 * log LMD).
So an MoM, using a fast method, will eventually beat FDTD or FEM. There are three basic types of fast methods:
1.) multipole methods (e.g. FMM)
2.) hierarchal matrices (e.g. ACA)
3.) pre-corrected FFT
I know Feko has a FMM code and Ansoft Designer uses a hierarchal matrix technique. I don't know of any other commercial codes that use fast methods, but I have not looked. If there are others, I am sure their users or other representative, will chime in.