This one gave me a laughing fit. About the only thing the Spartans and the Romans had in common was that they both wore swords and sandals.
Much of how the Romans handled combat was their own invention. Their key innovation was army organizaton from the legion down to the 100 man century (hence the word "centurion"). It manifested on the field as their unique form of phalanx and sheild wall fighting which was only made possible by their methods of group training. The Spartans for all their discipline fought much like the other Greeks did... only they were for the most part better at it than most of the Greek city states.
But despite their iron discipline, Athens beat the pants off of them every time they squared off.
Actually, Athens hid behind their walls in the Peloponnesian Wars and suffered a great plague as a result of the fact that they were periodically besieged by Spartans burning their crops.
The Athenians, to my understanding, actually never won a direct phalanx-to-phalanx engagement versus a force of Spartans. They relied on diplomacy and their navy to reach strategic objectives. The Spartans were always the premier land power of Ancient Greece.
If you're thinking of an army that defeated the Spartans in a direct engagement, perhaps you're thinking of the Sacred Band of Thebes, which was actually organized in much the same way as the Spartan military, down to the last brotherly-loving detail.
The Romans used a Phalanx, and it was different, that is true, from the Greeks, but fundamentally a similar construction. There's actually a fabulous discussion on this in a book by John Keegan about the history of ancient warfare, and the indication is that the Roman legion was superior to the Greek phalanx to the effect that it afforded them greater mobility in their smaller units. But in my own estimation, it was a tightly compacted group of men, shields drawn over their comrades, bristling with spears (prior to the introduction of the Gladius, that is). The proto-legion of the early Roman army probably would have been very similar to its Greek counterpart, especially considering their proximity and cultural origins to the Etruscans of Italy, a Greek colony.
But if it makes you feel any better, the Romans consistently lost to the Persians (the Parthian empire), whereas the Greeks held off an invasion of Greece.
Oh, and the Macedonian phalanx, which was derived from the Greek phalanx, conquered not just Persia but the entire mid-east, something the Romans could never do (because mid-east combat tactics were superior until the Byzantines came along).