While one is a space opera and the other crime epic, both introduced an innovative approach to narrative. The original Star Wars trilogy were the middle chapters so any new viewer watching A New Hope must orient themselves to the idea they're in a much larger story, one where the history and future are only suggested. The prequel trilogy (1999-2005) tells the origin story of Darth Vader and traces the decline of the Republic, while the sequel trilogy (2015-2019) serves as an epilogue/coda to the themes explored in the previous six.
The Godfather (based on the 1969 Mario Puzo novel) begins in 1945 and ends in the early 50s. The Godfather Part II covers two separate time periods: the origin story of the Corleone patriarch Vito spanning the early 20th Century and the story of Vito's son Michael taking over the family business. The Godfather Part III released in 1990 moves the story into the 1970s, providing closure to Michael's narrative. An as yet unmade fourth chapter would've continued the Corleone saga into the 1990s and provided another prequel story set in the 1930s centering on Sonny Corleone (covered in Edward Falco's 2012 novel The Family Corleone).
In 1977 Coppola edited the first two films together for chronological for their network TV debut. It's a different way to experience the films, but takes away from the time bending narrative of Part II. But one can get into The Godfather saga at any point and still be engaged with the characters. Like Lucas would do with his films, Coppola included new scenes left out of the theatrical release - speaking to how these narratives are continually evolving.
Now that all nine Star Wars films are complete, one can experience them in any number of ways. You can go in chronological order from episodes I-IX or watch them in order of release or countless other ways depending on your fancy. I prefer to watch them as they were released to get the authentic experience, but like The Godfather one can jump in at any point. The narrative inconsistencies in Star Wars I try not to get too hung up on. For example, originally Darth Vader was not to be Luke's father, and Luke and Leia were not siblings. As a fan it's easy to be confused due to Lucas's contradictory statements over the years, but the experience of the nine films transcends these minor narrative quibbles.
Narrative inconsistencies aside, Star Wars and The Godfather both center on families. The multi-generational approach feels more literary as we watch our characters age and pass on their knowledge (and issues) to the next generation. The Corleone family is a metaphor of the American experience: an immigrant family in America, building a future, and eventual corruption and downfall. The Skywalker saga begins with Luke, destined to become a messianic figure, while the prequels tell the story of his father Anakin. For the sequel trilogy the Skywalkers are less front and center as Luke, Han, and Leia pass on their legacy to the new generation. Both end with a sense of closure and continuity.
The sequel trilogy reminded me of The Godfather Part III in some indirect ways, especially The Rise of Skywalker. On the DVD commentary for Part III Coppola spoke of how he had little new to say in the last chapter since the audience knows the routine and what to expect. Star Wars movies are also recognizable to the point of ritual. Everything from the frequent quote "I have a bad feeling about this" to the opening crawl. The Rise of Skywalker delivered on all these points, despite the unbearable weight of history bearing down on the story. As the latter chapters suggest these stories can lose momentum, but also open the way for new stories told by others.
Star Wars and The Godfather on the whole are more than the sum of their parts. There are highs and lows in these stories, but it's about the journey, not the destination. They paved the way for a new type of story telling, tapping in the collective unconscious. The stories invite people in, which may be their most definitive legacies.