The phalanx did include shields (heavy round shields) not just long spears. Likewise the Romans also used spears and round shields at points in their history. The romans altered their tactics and equipment over time-the most 'common' visual image of a roman soldier with a long rectangular shield and a throwing spear comes from the later periods, long after the Spartians were gone.
My guess is that the Roman fasination for the Spartains probably stems from having their backsides kicked by them in battle. The Romans didn't win every battle against the Greeks, I think the Spartains were one of the hardest of the greek city states to subjue. Plus the Romans had a general interest in Greek culture and history that wasn't just confined to the Spartains-Athens was also held in high regard, but for different reasons.
lol, what?
Romans didn't build their tactics on phalanx, they built them on the weak points of phalanx warfare, which were the lack of mobility, archaic equipment, the inability to accept King Phillip's(of Makedonia) phalanx reform(hammer and anvil), bad adaptability and that it was almost strictly a defensive tactics. Phalanx tactics were supposed to the most powerful form of warfare after Alexander's conquest of almost all known civilized world(330 bc), but the truth was that once Romans reached Greece(around 120 bc IIRC), the role of cavalry had been vastly underestimated and almost dropped, so a regular tactic was to just roll out and spike the enemy to death while the odd cannon fodder ranged infantry(I think they were called pyloi) did what they could. Romans used a heavily ordered manipula formation to force hoplitai and phalangitai formations to broke ranks and superior technology to crush the poor lamellar-encased supposed "heavy troops", sweltering in their archaic hoplite gear. When Rome reached Sparta, the conquest didn't differ all that much. Although the exact history passages escape me.
Before actually entering Greece, Rome had been under Greek influence ever since the city was built, long before they formed an actual identity or conquered anything. Italy had several Hellenic city states(like Tarento and Croton for instance) and it was from their influetic trade and culture that Rome gained their first vestiges of power. Sparta and Athens weren't nearly as influental as those Italian states, but they did play a part in arts and such. Not to mention being important as trendsetters after a while. Of course Romans idolised Spartans for their rigour, everyone did, but they saw them like immortal heroes, inspired by thei discipline, but unamoured by their tactics. First Roman armies actually resembled phalanx warfare very much, however, and Italian phalanx wasn't a rare sight. It was as ubiquitous there as in Megale Hellas, but it didn't take that long for Romans to form their own tactics and equipment. Philip's reform did reach Italia as well, but the actual battles, or accounts of them, between Roman and strict Successor(of Alexander) armies in Italy were almost nonexistant. Pyrrhus, king of Epeiros, did use Successor tactics(bolstered with his elephants) and he did really well, considering everyone else had bitten the dust in Italy, but he only won that one infamous battle. Then it was all a downfall for the Hellenic states. And the rest of the world, hehhee.
You mentioned the "roman soldier with a long rectangular shield and a throwing spear", and that was actually not such a late invention, they just didn't form all of the base infantry until the Marian reforms around 1st century bc. Testudo on the other hand is almost a misconception, it wasn't nearly as overused as everyone seems to think and it wasn't even devised until near 1st century ad. The roman manipuli however, did resemble phalanx in the way that it was formed of ordered ranks of infantry side by side, advancing shields in front. That's where the similarities end, however, since the manipuli was a construction of mobile cohorts in checkerboard formation that could step back from the frontline and do all kinds of nasty maneuvers, while the phalanx was locked in combat the second they hit the enemy. The later form of shieldwall used widely in all of Roman Europe is actually what most people are after when they think Roman warfare. This was actually nicely portrayed in 300 too. The basic principle was that the frontlines would lock shields and stab each other to death over them. Very powerful, but the Romans had no need for it until nearly the end of the empire. Manipula was just too good, although it was dependent on the discipline of the armies in Republican and Imperial eras, discipline that the later imitations of those famous legions(Ceasar, Trajanus and Aurelius anyone?) lacked. Cue the Gothic horde in Rome.
To not base this whole thing on a strawhouse about to burst in flames(I haven't actually checked this stuff in ages), I suggest reading up on all this, it's interesting and gives you a nice perspective to ancient legends and mythic battles as well.
Man, can I ramble.