When I was in high school, I used to program two dimensional solar system simulations for fun (as you can surely tell, I was extremely popular and desired by all the ladies far and wide). I would put a "sun" in the middle of the screen and I'd spin "planets" around in simulated orbits.
There are essentially two ways you can go about programming solar system simulations. Technique 1, which for lack of a better term I will call the explicit technique, means deciding (or realizing) at the outset that the orbits of your planets will be elliptical, and then writing your program based on that. The explicit technique works because there are certain exploitable patterns in elliptical planetary orbits that can be harnessed to great advantage. Planets move faster when they are closer to the sun, for example, and slower when they are further away, in mathematically determined ways. One can derive formulae for the position of the planet on the orbit versus time.
Technique 2, which I used for my simulations and which for lack of a better term I will call the implicit technique, is at the same time simpler and more complex. You don't need to know anything about ellipses or how fast a planet moves when it's close to the sun or anything at all, in general, about how solar systems operate. What you do need to know are Newton's laws of gravity and motion and how to solve a first order differential equation with numerical techniques.
The implicit technique is simpler than the explicit technique in the same way that a lump of raw pork meat is simpler than a cooked pork sausage: the lump of raw meat is unprocessed and unrefined, while a sausage is both processed and refined. The equations of motion for a planet moving in a specifically elliptical orbit are derived from the more general and more generic laws of motion that come from Newton. Work and refinement are required to produce an equation which obviously describes movement along an ellipse, in the same way that work is required to produce a sausage from a lump of raw meat.
The implicit technique is also, paradoxically, more complex than the explicit technique because solving first order differential equations is more complex than just using equations which directly describe an elliptical orbit.
One key thing to note is that both methods produce the familiar elliptical orbits if you set them up correctly. The thing about the implicit method that makes it interesting is that you will get an elliptical orbit if the conditions are right, despite the fact that your program knows nothing at all about ellipses. Nowhere do you ever mention an ellipse in your program, nowhere are there any assumptions about how the orbit is supposed to look, or even that there should be a periodic orbit at all, but the ellipse nonetheless emerges from the simulation; it just appears naturally out of the math.
Which technique is superior? As always, it depends. If your only goal is to spin out a simulation of our own solar system, you can assume that the orbits are ellipses and your job will be easier if you use the explicit method. On the other hand, the implicit method is much more open-ended; once you have Newton's laws programmed into the simulation you can simulate a conventional solar system, but the same program will let you run simulations of binary star systems, systems with less or more planets than our own, systems that run normally until a heavy comet zooms by - pretty much anything. These sorts of fun and games are simply not possible with the explicit technique - these configurations were not built into the equations. To use the aforementioned analogy: once you have a lump of raw pork, you can make a sausage if you have the time and the inclination, but you're not limited to that. You can cut pork chops, carve out a fletch of bacon - anything you want. If you start with a sausage, all you get is a sausage.
The essential difference is that the implicit technique is less presumptuous about what to expect when the simulation runs. As I mentioned, we don't assume that the orbit will be an ellipse, or even that there will be an orbit at all. We just enter the laws of gravity and see what happens. The results are often surprising. Yes, I found that I could obtain the familiar elliptical orbit on my monitor, but I also found that, for example, if a planet is moving too fast, you don't get an orbit at all. Instead, the planet "slingshots" around the sun in a trajectory shaped like a parabola - something that was not obvious to me at all.
Since playing around with these programs I have learned two things:
the simplistic numerical method I "invented" to solve Newton's laws of motion in my simulation has actually been attributed to Leonard Euler and is dubbed "Euler's Method". It's not very good as the errors are quite significant unless you take an extremely small time interval.
more importantly, the elliptical orbits and parabolic trajectories that appeared "naturally out of the math" were examples of what I later learned were called "emergent properties". These are properties of the system as a whole, things that are not obvious until you actually get your hands dirty and try things out.
Emergent properties are basically macroscopic effects of local behavior. There are many examples of emergent properties in nature. One example is Jupiter's Great Red Spot, that enormous, stable yet chaotic storm that just doesn't seem to go away. Once scientists program the proper Jovian weather model into a computer, the Great Red Spot just emerges out of the calculations. It's not something that's obvious from the math, and the program doesn't know anything about Great Red Spots or storms, but it's nonetheless there, in the model, in a fundamental sense.