Many times, the equations to be solved represent a physical conservation law. For example, the Euler equations for fluid dynamics are representations of conservation of mass, momentum, and energy. Given that the underlying reality that we are modeling is conservative, it is advantageous to choose methods that are also conservative
You can also see something similar with electromagnetic fields. Maxwell's laws include the divergence-free condition for the magnetic field, but that equation is not always used for the evolution of the fields. A method that conserves this condition (for example: constrained transport) helps match the physics of reality.
@hardmath pointed out that I forgot to address the "what could go wrong" part of the question (Thanks!). The question specifically refers to engineers, but I'll provide a few examples from my own field (astrophysics) and hope that they help illustrate the ideas enough to generalize to what might go wrong in an engineering application.
(1) When simulating a supernova, you have fluid dynamics linked to a nuclear reaction network (and other physics, but we'll ignore that). Many nuclear reactions depend strongly on the temperature, which (to a first-order approximation) is some measure of the energy. If you fail to conserve energy, your temperature will be either too high (in which case your reactions run much too fast and you introduce far more energy and you get a runaway that shouldn't exist) or too low (in which case your reactions run much too slow and you can't power a supernova).
(2) When simulating binary stars, you need to recast the momentum equation to be conservation of angular momentum. If you fail to conserve angular momentum, then your stars cannot orbit each other correctly. If they gain extra angular momentum, they separate and stop interacting correctly. If the lose angular momentum, they crash into each other. Similar issues occur when simulating stellar disks. Conservation of (linear) momentum is desirable, because the laws of physics conserve linear momentum, but sometimes you have to abandon linear momentum and conserve angular momentum because that is more important to the problem at hand.
I have to admit, despite citing the divergence-free condition of magnetic fields, I'm not as knowledgeable there. Failure to maintain the divergence-free condition can generate magnetic monopoles (which we have no evidence of currently), but I don't have any good examples offhand of issues that might cause in a simulation.