What is the difference between Mode Matching software and other 3D solvers?

In short: The method of solving Maxwell's equations.

Mode-Matching (MM) method + derivations: The electro-magnetic fields inside the structure under consideration are expanded into known, analytic solutions of Maxwell's equations for similar geometries. The boundary conditions are fulfilled using a Galerkin procedure on the element surface. In the normal case only a few of these (some 100) analytic solutions (modes) are needed for sufficient convergence of the results, so this method is very fast and has low computational requirements. Because, there are only few structures with known, analytic solutions of Maxwell's equations (e.g. rectangular + circular cavities + waveguides), this method is applicable only to certain geometries and thus lacks some geometric modeling capabilities (inclusion of screws, rounded corners, draft angles, etc.)

3D solvers (FEM and FDTD): The electro-magnetic fields are expanded with very simple, locally defined expansion functions (e.g. edge element functions in FEM, delta functions in FD). Both Maxwell's equations and the boundary conditions are fulfilled either via functional (variational) analysis (FEM) or approximation of the differential operators with finite difference operators (FD). Depending on the size (in wavelengths) and the geometric complexity of the structure you may need a lot (10^4 ... 10^6) of expansion functions to achieve sufficient convergence.

Although the systems of equations are sparse (few non-zero elements pre row/column), you'll need either a lot CPU time (iterative solvers) or lot of RAM (direct solvers) of both. The 3D solvers have the advantage to offer full geometric modeling capabilities including all features (e.g. screws, rounded corners, draft angles) which are impossible or difficult to include into the MM method.

In the µWave Wizard each element can be analyzed with its own, most advantageous method. So there's no need to analyze the whole structure at once with a 3D solver. You can mix 3D solver (FEM) elements with Mode-matching elements in the same structure. Both element types have a common multi-modal interface, so they are fully compatible and interchangeable. With the breakdown of the structure into smaller elements you even profit when all parts are analyzed using a 3D solver, since the solution of many small systems of equations is still faster than solving on single big system of equations.