Rome prevailed over her neighbors and adversaries mainly because of several factors:
1) In the first stages of her rise, Rome was uncommonly generous in granting her citizenship, or at the very least Ius Larinum, to conquered people. As historian T. J. Cornell said “ The Roman Republic was like a criminal racket which subjected cities were made part of”. Rome shared the spoils of her victories with her allies, granted them citizenship in established colonies or in their own cities, and all she asked was military support, no tribute at all. Rome expected to fund herself with plunder conquered by the enemy, and she always did. Of course, Rome had to prove herself strong enough to work in everybody’s interest, and that brings in the second point.
This does not seem particularly different from what other Italic polities did, or would have evolved into doing if not absorbed by the Roman behemoth. It may be that Rome was, from early times, even more liberal in this reard than the Italic standard, due to its possibly more-multiethnic-than-usual formation, but I think that we don't know enbough detail about the rest of Italy to be certain. Clearly what they did in this regard was total anathema to the Hellenic mindset (and presumably exceptional in the Punic world). Also, the political system of Rome looks like to be a bit more inclusive the what was the norm for Etruscan city states, though of course this was the product of a long and bitter internal class struggle. Similar struggles occurred in Etruria of course, but it seems that th aristocracy there won hands down. In Rome, the yeomen class kept, and then expanded, a political say, that then made them willing to form the formidable army we all know.
2) The Roman army was the most effective military organization of ancient times. Its tactics were simple, but ingenious. Just three lines of heavy infantry, each divided in 10 maniples, the gap between maniples of the same line filled by those of the line behind. Not only it was incredibly mobile, but allowed a Roman general to deploy fresh reserves after fresh reserves, which would eventually break the tired enemy line and bring it to flight. Before Hannibal, a genius of tactics, Rome had ever suffered only a handful of serious landlosses, and it was either because the soldiers were trapped (Caudine Forks), because they were faced with a previously unknown military device (Pyrrhus’ elephants) or because they had very little cavalry (Battle of Tunis). The battle of Allia doesn’t count, for all we know, it might’ve been just a skirmish gone wrong. In any regular battle on the field, Rome trounced her enemies.
Probably Allia is also too early. The exceptional Roman war machine was truly honed in the Latin and Samnite wars; Livy states clearly that the manipular order was adopted around the time of Caudium IIRC. Of course, the Roman army before was already as good as anything else in Italy, but not in a different league.
3) Cohesion. One thing peculiar of Rome, never in her life she was member of a federation of cities. The Etruscans, the Samnites, and, for what we know, all Italic people were organized in federations whose members didn’t always share a common interest, nor they were particularly cohesive. Very often, some Latin cities would submit and ally to Rome, while others would revolt. In Rome’s case, every single one of her cities acted according to Rome’s will, never indipendently, which also allowed for better organized military operations.
Rome was part of a federation (the Latin League). It simply managed to outgrow the rest of it to the point to take it over fully. The overall cohesion of the Roman Italian hegemony is undisputable, but it is the product of a long evolution that was not an inherently necessary development at the start.
4) Patriotism. There were several factions in Rome, as in all other ancient cities, but differently from those, the plebs or the patricians never, ever considered for one second calling in a foreign army to settle their business. What happened in Rome stayed in Rome, no Roman citizen would have dared endanger Rome in such a way. The only two people in the entirety of her history who led foreign armies against her were Sertorius and Quintus Labienus, and, at least in the former’s case, not to the total benefit of the foreigners.
This is hardly unique in concept (though perhaps it is nearly so in intensity; but look at how steadfast the Carthaginians proved to be 146 BCE), and also the result of a long evolution (including a specific religious dimension). The Tarquinii had no qualms about bringing foreign armies to retake power, for example. Again, the endgame is undisputable; but I would say that it was a result of successful imperialism (made possible and caused by a cohesive internal social structure whereby all the elites and to a lesser extent the common people reaped the benefits of such cohesion) more than its root cause.
5) Ruthlessness. If Rome went to war, there was only one acceptable outcome: crushing victory. If the war did end in a draw, it meant that Rome was too caught up in other wars, and would soon go back to finish the job. Pyrrhus and Hannibal thought their victories would be enough to submit Rome, and had it been any other city, they would have been right, but Rome was something else entirely.
This seems to be case only after the late fourth/early third century BCE, and the combined effect of some serious victory disease that happened to keep reaping victories nonetheless.
You want Rome to fail? Take away her army and she will, despite everything.
With the important specification that, at least from mid-fourth until about mid second century BCE, to take away Rome's army more or less amounts to take away Rome's society as a whole.
