Star Wars is Greek; Star Trek is Roman. 

The comments have very valid points, and if you see this reblog, you should read them as well, but the initial idea was that Star Wars has a lot of focus on individual heroism - there are teams saving the day, but in the end the OT is centered on Luke’s personal struggle with himself and the Dark Side, and there’s much more of the sense that a single person can affect the galaxy with wits and/or a lightsaber - it’s more Iliadic or Odyssean. 

Star Trek, on the other hand, is about crew cohesion and bringing the ideals and material benefits of the Federation to other worlds, and serving the Federation or your crew above yourself, which feels like the ideals of Republican Rome - you’re not going to get Scaevola, sacrificing his hand, not for personal ideals but to show the virtues of his people, in Star Wars, but you might in Trek. 

Every ship captain we see in Star Trek (but especially Picard) is a philosopher-king, while Star Wars has the fall of the Republic and the rise of the Empire.

I guess you can go either way with this question, depending on whether you’re comparing the series to Greek vs. Roman art or Greek vs. Roman politics.