This is actually a pretty complicated question to answer and a couple of points have to be clarified up front. For one thing, it’s easy to think of Rome as a kind of monolithic body that stood through the centuries, but in fact we’re talking about a very long period of time that saw many dramatic changes occur in the political, cultural, economic, and military spheres. While there are certainly threads that carry through the centuries in all these respects and others, it’s important to note that even at an organizational level things were constantly in flux – the military is no exception.

The broader point to keep in mind is that when you say “organized structure,” you’re either subconsiously or consciously evoking the popular conception of Rome as organizational/logistical/engineering masters who subjugated their less developed or determined neighbors with brutal discipline and efficiency. The problem with this thinking as it relates to your question is that it doesn’t take into consideration the fluidity from decade to decade regarding the societal components I mentioned above, and the fact is that these in many ways play as important a role in determining Roman military success or failure as does sheer discipline or tactics.

Let me give you an example of what I mean. The popular idea of the “fall of Rome” is that hordes of big scary barbarians swarmed down from Germany laying waste to all before them. In fact, the barbarian invaders could not in any sense ever hope to knock the empire down in that way (they actually didn’t even want to do that but that’s a different point). The Roman army was huge and highly developed at this time. They outnumbered the invaders by vast margins and had much greater skill and technology and organizational tradition. So if this is true, why did they let the invaders through? One reason is that the political body which fed the military and provided the framework through which it breathed was rotting and starting to disintigrate. All of those forts that ran along the Rhine were being abandoned one by one as various civil wars siphoned soldiers away, and then lacked the social stability to reestablish order. In other words, despite organizational superiortiy, they still lost big time.

Consider too that their military success was by no means inevitable. Read Livy’s account of the second punic war and you’ll see what I mean. That war was a titanic life or death struggle and when Rome finally snuck out on top, it wasn’t because they were better organized or disciplined. To this point, it’s more useful to examine each war individually and understand what was going on internally and externally that resulted in either Rome’s victory like the battle of Zama for example, or defeat a little over a decade previous at Cannae.

In summation, the question to ask yourself when thinking about these things is “what made Rome win/lose over X at this point in time” rather than trying to find one single factor that gave them an edge. -- There are examples by the way where organizational sophistication were certainly a major contributing factor to victory in the way you imply. A good example would be Caesar’s conquest in Gaul.

--- One quick side note about what we know of Roman military organization itself, which is, not as much as we think. We get the heirarchy all right, but despite popular images that depict the legions lined up and replacing one another in waves (the HBO series Rome comes to mind with the Centurion whistling to switch the lines), there is no actual evidence that this ever happened. Livy and Polybius both mention something similar to this happening, but they differ in points. We also for example don’t know how exactly Caesar fought in Spain. He mentions in his commentaries that he had to mix up his strategy b/c the Pompeians ‘fought like the natives’ but what this looks like in practical terms is lost to us.

Please refine your question. What do you mean by "organized structure?" The chain of command? The division into legions? The career path for the soldiers? The battle field tactics? What?

I'd like to point out that the Roman military went through many different iterations in command structure, divisional structure, recruitment techniques, funding sources, tactics, and equipment.

In general, the Roman military was highly effective due to its extreme regimentation and a cultural disposition toward an almost unnatural sense of the "honor" and "glory" of serving. The Roman military (both naval and land arms) often succeeded through its engineering capabilities and drive to acquire new technology.

I think at all stages however, the Romans' greatest asset was the vast supplies of men and money. The Romans lost plenty of battles, even entire armies with both heads of state and whole navies, against Carthage and had devastating reverses against Pyrrhus; they, however, were able to raise new armies almost immediately, generally through conscription and forced contribution to the war chest (in early republican times). I believe this sheer volume of resources was the most important factor, although the Roman genius for organization, command structure, and engineering were certainly huge contributors to the success of Rome.