Computational Modeling vs. Mathematical Modeling

Just recently been thinking about “computational modeling”, and I’ve been thinking more and more that the important thing is not so much the “computational” part of modeling (the fact that you’ve simulated a model on a computer), but rather the mathematical model that is so implemented.  I think that ultimately we’re interested in having models of how things work.  In this case, I think most of the “understanding” is in the models themselves, as stated mathematically, not in a particular implementation of the model.  Yes, computational implementation is important because: 1) it keeps us honest by forcing us to make our models physically realizable (and therefore explicit and consistent), and 2) it can reveal aspects of our model that might not be apparent to us initially.  But my sense is that the mathematical model is more important than its “computational” realization.