The universality of Mathematica does not mean that it solves each special problem instantly. If you have a system of coupled tricky reaction-convection-diffusion PDEs you might need a tricky solver that enhances, say, finite element techniques. We have such animals in advanced quant finance, processes in chemical reactors have, ....
But,
Mathematica does so many things FOR YOU that I would need far too many pages to describe. You sit down and compute symbolically and numerically, visualize ....
Mathematica offer to do infinite many things WITH YOU by enabling you to make programs out of your computation in an unprecedented intuitive declarative manner and a seamless workflow ...
and
You can do things FOR MATHEMATICA (close to perfection in version 8) by integrating your own specialized solvers into it. Doing this you extend the Mathematica kernel and your special objects speak Mathematica. Making it a package and offering it to the market you extend Mathematica into a special domain universe.
Like UnRisk extends Mathematica into the universe of advanced financial derivatives and risk analytics - the UnRiskverse - instantly used by traders, treasurers, asset and investment managers, fund managers, risk controllers and risk managers and programmatically extended by quant developers in banks, capital management firms, funds and insurance world-wide.
That can trigger innovative spirals you cannot drive in any other environment (to my knowledge).
