"They spoke the same language" ... not exactly, the Spartans spoke Doric, while the Athenians Attic. (Interestingly, there is a few Doric speakers left [0].) While those languages were related, their mutual intelligibility was limited. Instead of "Greek" as a single language, you need to treat it as a family of languages, like "Slavic".
"shared the same literature" ... famously, the Spartans weren't much into culture and art, and they left barely any written records of their own. Even the contemporaries commented on just how boring Sparta was in all regards.
If we delve deeper into ideas about how a good citizen looked like, or how law worked, the differences between Sparta and Athens are significant, if not outright massive.
While those two cities weren't entirely alien to each other, had some ties, same gods, and occassionally fought on the same side in a big war, there was indeed a huge political and cultural distance between them. I would compare it to Poland vs. Russia.
