Dr.D said:
The point of my story is just this: Both the physicist and the engineer would like to solve the problem, but when it gets messy, the physicist often seems to be inclined to say, "It has a solution, but it is not worth the effort to actually find in most cases. We will only work the simple special cases."
The engineer, on the other hand, is much more inclined to say, I have to have the solution for all cases, so even though it is messy, I will resort to a numerical solution as a matter of routine.
Now I am well aware of the fact that in research, physicists often do use numerical methods of solution. But it is in the undergraduate teaching area that I have never seen it happen, whereas it is fairly routine to apply numerical methods in undergraduate engineering problems. To return to your question, this is the significant mathematical difference that I see between Engineering and Physics education.
You characterization of the professions there seems a little disingenuous, and your qualification at the end doesn't really reverse that impression.
Though in some sense a physicist might not "put out the fire", you seem to say it's because it's too messy and complicated for them, whereas engineers are much more willing to get their hands dirty. Rather, any competant physicist should be able, and willing, to get messy to solve the problem if necessary; the reason they may not is that solving the problem in full detail is simply not what they are interested in - they want the core physics of the situation, not every detail.
The 'core physics' is often are there regardless of how complicated the system is, hence it is seldom enlightening to model the whole system when a simplified model captures the essential physics one is interested in. All the complexities of the system could even obscure the core physics one is trying to look at. Take Newton's first law being obscured by the presence of friction, for example.
An engineer, on the other hand, is much more interested in the practical application of the physics she knows, and to that end solving the problem in full detail is necessary to produce a real version of the system. Similarly, any experimental physicist who needs to model some appartus in order to calibrate it, etc, is going to model the hell out of it - they won't simplify a damn thing.
So, although your 'story' may in some sense be an accurate characiture of physicists, mathematicians and engineers, it's important to stress WHY their actions are different - because what they're fundamentally interested in aren't the same for each - and I don't think you did that with your post, so it was somewhat misleading for someone like the OP.
As for why solving things numerically isn't done in undergrad - well, I suppose to some extent it's again because they're interested in different things. It's also probably somewhat due to the confidence of physicists that they're clever enough to figure out how to do something if they really need to. I've never taken a computational physics class so I don't know what sorts of things are taught in one, or what detail is achieved. I've certainly done coding in undergrad for various classes that involved various levels of modelling things - perhaps not quite at the level you're envisioning, though.