A different system of government would make Sparta, well, not-Sparta.
It would change the whole dynamic of Greek history, as a likely result would be one of the following:
a) Sparta runs through the cycles of regime changes, tyrannies and civil wars experienced almost anywhere else in the Greek world.
b) Sparta remains a relatively stable, backward and not particularly noteworthy polity in its part of Greece; its lacks it legendary military prowess and any ability to dominate the Peloponnese as a whole.
In either case, the great geopolitical and ideological confrontation with Athens does not happen. Athenian hegemony would still be attempted, and challenged by other cities (this remains fairly likely even if Athens does not develop its particular democratic institution, which indeed it may not in this scenario, given the historical involvement of the Spartans in the ousting of the Pisistratid tyrants). But, even assuming minimal changes on the Athenian side, the most likely challengers would then be less strictly oligarchic, mercantile cities whose interest would clash with Athenian ones without making the conflict an ideological existential struggle. This would make the Greek civilization as we know it completely different: nothing like Plato's philosophy would ever arise, for instance.
This, disregarding the butterflies involving the big, big, big empire on the Asian side of the Hellespontus.