If I rember correctly from a history channel episode they wore heavy bronze armor, not fighting shirtless. While other greek hopilites wore a different type of armor
Also wasnt the tesudo(srry if misspelled) not a fighting formation as the Greek Phalax was but more used to protect Roman solders from enimes archers.
Finally I dont belive the Romans every really fought Spartans from the city-State Spartan (like in 300) but fought the unified Greece under Macedon rule. Macedonian rule over all of Greece came under Alex the Great
Remember, we're talking about the early Spartans around 400 BC, even earlier at this time such armour like bronze wasnt used and pretty much mass produced as with the Romans or Greeks for that matter. The Spartans usually dressed light, as wearing such heavy armour would be quite restrictive in combat and with the Phalanx formation already as it is being so restrictive it wouldnt really help. As I have said the Spartans traveled light, as most of their fighting was just to defend and they werent used as an offensive force as the Romans. The heaviest thing a Spartan would have had is probably his shield, the Spartans were fond of their large round shields and certainly defended himself and the man to the left of him well.
Macedon didnt rule Greece when the Romans had came, After Alexander the Greeks divided up into the Thracian empire, Macedonian, Greek and Seleucid empires.