Analytic methods: almost always approximations. Very few exact solutions. Good for understanding basic physics, but often break down in regions of interest such as phase transitions. Little knowledge or control of errors, not easily systematically improvable.
Monte Carlo methods: calculations usually involve sums or integrals over very large number of dimensions or configurations. Monte Carlo is the obvious choice. Can compute quantities for any parameter values (including where analytic methods don't work). Errors can be estimated, and are systematically improvable by using more sample configurations and larger systems (i.e. more computer power!).