Greetings!
Hmmm. Well, to be honest, neither one is necessarily "better" than the other, as far as campaign settings go. Both Ancient Greece, and Ancient Rome can readily accomodate the exact same campaign elements--each particular setting however, does it in a different way. That particular way will simply be preferable based on how the DM wants to go about running the campaign, and what areas and styles to emphasize.
Ancient Greece:
Whether you are running a campaign set from the ancient Dorian barbarian invasions and the destruction of Mycenae, to the storming of the walls of Troy, or the great Pelloponesian War, the various conflicts between the city-states of Sparta, Athens, among others, to Alexander The Great's sacking of Persepolis and the annihilation of the world's largest and most powerful army, that of the Persian Empire, and the subsequent march to the great Indus River, and the reaches of the jungles of India, and even to the wars of the Successor States--Ptolomy, Seluicid, and so on provide plenty of exotic cities, vast deserts, political intrigue, seduction, palaces full of women, slaves, assassins, royal marriages, espionage, and glorious treasures to keep a campaign going for years.
Of course, Ancient Greece, as well as the periods stretching through Classical Greece, to the Hellenistic Age, offers outstanding exploration milieu's, gods and demi-gods, great cities, ferocious monsters and wondrous adventure.
Ancient Rome:
Ancient Rome, offers everything virtually that Ancient Greece offers, simply through different style and method of delivery. Ancient Rome, from the early days of foundation, all the way through to the republican years of Marius, offers savage frontiers of barbarian invasions, monsters on the frontiers, and lots of city adventures, politics, and intrigue.
Shifting a bit in focus, to the time of Caesar, well, again, scope for city intrigues, gladiatorial games, circus races, barbarian invasions, monsters on the frontiers, rebellions, assassination plots, and glorious armies marching to years and years of bloody war, glorious conquest, and empire building seldom seen in scale through all of history.
Periods later on in Ancient Rome--whether focusing on the period of Imperial Acsension, the time of the Five Good Emperors, or imperial decline, the era of Imperial Resurgence under Diocletian, Justinian, among others, until the fading years of the empire's glory when corruption, war, among other changes had sapped the Roman Empire's ability to heal and regenerate itself, and ultimately to rise again to a period of glory and dominance.
Generally speaking, both campaign settings offer vast scope for the same kinds of adventures. Whether urban based, wilderness or frontier-based, army-focused, political/intrigue, exploration/discovery--both settings can do it all, and in spades. Ancient Rome simply offers a bit higher in the technological aspect of things, a tighter political environment, and a bigger campaign scope.
Semper Fidelis,
SHARK