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Analytical means you're finding an exact solution using equations ~ 1+x = 5, solve for x. Numerical means that you're using an approximation method to find the solutions ~ like finite difference time domain method or something.
Practically, analytical solutions are things you can find for very simple and/or specific problems (IE involving perfect spheres/cylinders, infinite ground planes, etc). Numerical solutions are approximations useful when you can't simplify a problem enough, like in most real problems. Simulators using FDTD, FEM, etc are approximations. Numerical approximations are generally ones made to make it possible to simulate by computer.