What I meant was importance on a symbolic level: spears clearly were used as symbols by the various ancient incarnations of the Roman state, swords, however, not so much.
I think it's because spears were a symbol of war, and, therefore, of military might. In Herodotus there is a line by Leonidas who tells a subject of Xerxes, "If you knew freedom, you would fight for it not just with spears, but even with axes". Spears here are the military weapon of choice, while axes are civilian items used as improvised weapons in an imaginary desperate and underequipped revolt. Swords would have been less interesting to name, because they were somewhere in between the extremes: military weapons that weren't as apt or as important as spears. Spears vs swords is like rifle vs pistols, a state will more likely exalt a soldier's rifle than a pistol.
I would add another detail, which is the severe arms control in the republics of the ancient era: even when you were allowed to own weapons, you generally couldn't carry them in public places. What these places were varies by time and place; Rome had the pomerium, other places had the temple, the market, and the agora, or the whole of the city within its walls. So a sword couldn't become a status symbol like in later eras, because you didn't wear one when performing your public duties.
Instead, two civilian weapons gained special regard in Rome: sticks and axes, that represented the power of the magistrate to punish citizens with beatings and decapitations and were bundled in the fasces. The lictors carrying them weren't there just for show, but actual jailors, torturers, and executioners that accompanied the magistrate to enforce his decisions.
In medieval times, the Doge of Venice occasionally carried a bare sword for similar reasons while parading someone sentenced to death. Swords represented the might to punish evil deeds. For similar reasons, we see as Justice carrying a sword. In Greece, however, the goddess of justice (Dike) had many aspects and identifications, and so she could show a number of implements, among which the cornucopia and lighting (as Astraea), parts of a mouse trap or a mallet (to beat up Injustice), a sword (as an infernal deity), a box full of books and more.
Just for clarity, not knowing if your "chuck" meant "cast aside" or "throw", I don't believe the Greeks in this age threw their spears before clashing with the enemy. There is a description in Herodotus where the Spartans have turned to swords because they have to, as they have been fighting for so long that most of their spears had been broken.
Interesting; legionaries could be rewarded instead with the hasta pura, a spear without the iron tip (different authors give different occasions when it was bestowed; one was the retiring of centurions holding a particularly important position).As for Roman use of swords as symbols, you do see them in the form of wooden swords/Rudis given to retiring gladiators. Not a military association, but a martial one at least.
This sort of purely symbolic rewards (not much worth to them) was to be found in the panellenic games, too: a laurel wreath or branch in Olympia and Delphi, a celery one at the Nemean games, and a pine one at the Isthmic games.
Gladiatory games, suspended between battle and civilian life, do look like one of those places where a sword could gain a very special meaning.