Athens and Sparta were two very different city-states. Here are some of their main differences.
1) Government structure- Sparta was a military state, meaning their military controlled everything. Citizens did not have the ability to vote and males aged 7-30 were forced to serve in the military. Athens, however, had a democracy in which citizens go to vote on changes made in their society.
2) Art/Culture- Sparta's militaristic society left little from for individuals to study academic subjects. However, Athens became a center for educational thought and reasoning. Topics we study today like philosophy, mathematics, and science have all been affected by the finding of citizens from Athens.