The Graeco-Roman world is a unique example of intertwined cultures, the geographical and historical proximity of the two civilizations is such that's it's often impossible to distinguish where the one ends and the other begins. In extremely broad terms, it wouldn't be inaccurate to say that the political system of the Romans were heavily influenced by the various political systems of the Greek world, after all the high point of the Greek world is earlier than the high point of the Romans, it's only natural that the Romans would benefit from the political philosophy of the Greeks.
Livy's story may very well be apocryphal, he is separated from the events he describes by four centuries and in the preface of Ab Urbe Condita he notes:
Such traditions as belong to the time before the city was founded, or rather was presently to be founded, and are rather adorned with poetic legends than based upon trustworthy historical proofs, I purpose neither to affirm nor to refute. It is the privilege of antiquity to mingle divine things with human, and so to add dignity to the beginnings of cities; and if any people ought to be allowed to consecrate their origins and refer them to a divine source, so great is the military glory of the Roman People that when they profess that their Father and the Father of their Founder was none other than Mars, the nations of the earth may well submit to this also with as good a grace as they submit to Rome's dominion.
Source: The History of Rome, Book 1, preface, translated by Benjamin Oliver Foster
Nevertheless, Livy is not the only historian to claim Greek influences to the Roman political system. Both Polybius' Histories and Cicero's De re publica, our primary sources for the Roman Constitution are full with references to Greek influences. Polybius, an Arcadian that spend part of his life in Rome, is a much earlier source than Cicero and Livy, who were near contemporaries. Book IV of his Histories is an examination of the Roman Constitution, with direct comparisons to various Greek constitutions, and in it we find an assertion that the Roman Constitution was very similar to the Spartan one:
Lycurgus however established his constitution without the discipline of adversity, because he was able to foresee by the light of reason the course which events naturally take and the source from which they come. But though the Romans have arrived at the same result in framing their commonwealth, they have not done so by means of abstract reasoning, but through many struggles and difficulties, and by continually adopting reforms from knowledge gained in disaster. The result has been a constitution like that of Lycurgus, and the best of any existing in my time....
Source: Histories 6.10, Polybius, translated by Evelyn S. Shuckburg.
Cicero takes this a step further:
Lycurgus in Sparta, formed under the name of Geronts, or Senators, a small council consisting of twenty eight members only; to these he allotted the highest legislative authority, while the king held the highest dominative authority. Our Romans, emulating his example, and translating his terms, entitled those whom he had called Geronts, senators, which, as we have said, was done by Romulus in reference to the elect patricians. In this constitution, however, the power, the influence, and name of the king, will still be pre–eminent. You may distribute indeed, some show of power to the people, but you inflame them with the thirst of liberty by allowing them even the slightest taste of its sweetness, and still their hearts will be overcast with alarm, lest their king, as often happens, should become unjust. The prosperity of the people, therefore, can be little better than fragile, when placed at the disposal of any absolute monarch whatever, and subjected to his will and caprices.
Source: The Political Works of Marcus Tullius Cicero: Comprising his Treatise on the Commonwealth; and his Treatise on the Laws. Translated by Francis Barham.
De re publica is an excellent read, I strongly urge you to read the full text. Just for completeness sake, here's another quote that I think is a beautiful answer to your question:
Our Roman constitution, on the contrary, did not spring from the genius of an individual, but of many; and it was established, not in the lifetime of a man, but in the course of ages and centuries. For (added he) there never yet existed a genius so vast and comprehensive as to allow nothing to escape its attention, and all the geniuses in the world united in a single mind, could never, within the limits of a single life, exert a foresight sufficiently extensive to embrace and harmonize all, without the aid of experience and practice.
Ironically, the fact that the Romans were quite open to integrating ideas from other cultures is also one of the distinctive traits that separates them from the politics of classical Greece. The Roman policies of acculturation, integration and assimilation were an extremely important factor in sustaining their diverse empire, something that wasn't really a necessity for the Greek city states. We can only speculate, but I think you'll agree with me that if someone suggested offering citizenship to political allies in the Athenian assembly during the 5th century BC, they would be - at best - laughed at.
Returning to Livy, there are three notable flaws in his story of a Roman delegation to Athens:
- The similarities between the Law of the Twelve Tables and the Solonian constitution are limited and inconclusive.
- There's no mention of the Roman delegation from Athenian writers.
- A Roman visitor to Athens in 451 BC would have found Athens amidst a wave of political reforms, spearheaded by Pericles. Solon's constitution was, at the time, almost a century and a half old and undergoing its third major reform, had the Romans actually visited Athens they would have witnessed far more complicated procedures than the Solonian constitution.
A far more likely story is that the Romans learned of Solon's laws and the various other Greek political systems through their contact with the Greek colonies in Magna Graecia. Greek presence in Italy predates Rome's founding and according to legend the fifth King of Rome, Lucius Tarquinius Priscus, was of Corinthian descent. According to Livy, Lucius Tarquinius Priscus significantly altered the senate, doubling its membership to include the "lesser families":
Ancus reigned four and twenty years, a king inferior to none of his predecessors in the arts of peace and war and in the reputation they conferred. By this time his sons were nearly grown. Tarquinius was therefore all the more insistent in urging that the comitia should be held without delay to choose a king. When the meeting had been proclaimed, and the day drew near, he sent the boys away on a hunting expedition. Tarcuinius was the first, they say, to canvass votes for the kingship and to deliver a speech designed to win the favour of the commons. He pointed out that it; was no new thing he sought; he was not the first outsider to aim at the sovereignty in Rome —a thing which might have occasioned indignation and astonishment, —but the third. Tatius indeed, had been not merely an alien but an enemy when he was made king; while Numa was a stranger to the City, and, far from seeking the kingship, had actually been invited to come and take it. As for himself, he had no sooner become his own master than he had removed to Rome with his wife and all his property. For the greater part of that period of life during which men serve the state he had lived in Rome, and not in the city of his birth. Both in civil life and in war he had had no mean instructor — King Ancus himself had taught him Roman laws and Roman rites. In subordination and deference to the king he had vied, he said, with all his hearers; in generosity to his fellow-subjects he had emulated the king himself. Hearing him advance these not unwarranted claims, the people, with striking unanimity, named him king. The result was that the man, so admirable in all other respects, continued even after he had obtained the sovereignty to manifest the same spirit of intrigue which had governed him in seeking it; and being no less concerned to strengthen his own power than to enlarge the state, he added a hundred members to the senate, who were known thenceforward as Fathers of the “lesser families,” and formed a party of unwavering loyalty to the king, to whom they owed their admission to the Curia.
Source: The History of Rome, Book 1, chapter 35, translated by Benjamin Oliver Foster
All that said, and while influences from Greek political systems are both evident and reasonable throughout the history of Rome, saying that the Romans copied their political system from the Greeks is extremely inaccurate. At the core of Roman politics there's a unique dichotomy, the Senate and the People, exemplified in the Latin phrase Senatus Populusque Romanus that came to be the symbol of Rome.
The phrase may be much later, but the political dichotomy existed from the early days of the Roman Republic, and it's incompatible with the direct democracy of the Athenians. In extremely broad terms, the Roman political system is somewhere between the Spartan oligarchy and the Athenian democracy, influenced by both to some extend, but not directly copying any of them.