The relationship between the Marauders would be very different. In the original, James was dead, but if he's alive, they could continue their adventures together. And the relationship between the Potters and the Weasleys might also change. Maybe they would be closer allies from the start. Also, Snape's story would be completely different as his main motivation in the original was his love for Lily and his guilt over her death.
Well, if James and Lily are alive, Harry doesn't grow up an orphan. He has his parents to guide him. For instance, he might not go to Hogwarts with the same sense of longing for family as he does in the original. His character development would be different as he has a loving family environment.
In a fanfiction where Lily and James Potter are alive, the plot would be turned on its head. For starters, the Dursleys would never have come into the picture. Harry would have been raised in a loving magical family. This would mean he'd have access to all his magical knowledge and abilities from a young age without any suppression. Also, the war against Voldemort would have been fought very differently. Lily and James, being experienced and powerful wizards, would have led the charge. Their presence would have inspired a different kind of hope in the wizarding community. And think about Hogwarts - Harry would be going to school with his parents' support and influence, which could change how he interacts with his classmates and teachers.
They change it a great deal. For starters, Harry wouldn't be an orphan. He would grow up with his parents, which would completely change his personality. He might be more confident and less brooding. Also, the whole dynamic of the wizarding world would be different. There would be no need for Dumbledore to have as much control over Harry's life. James and Lily could have their own influence on how the wizarding world is run.
In some fanfics, James and Lily survive and start to work on a new form of magic that could be used to defeat Voldemort once and for all. They collaborate with the wizarding world's greatest minds, and their survival gives hope to everyone. This changes the mood of the story from one of loss and struggle to one of hope and determination.
The relationship dynamics change drastically. Instead of having that tragic but heroic backstory of his parents dying for him, he has a story of being left behind. This would also change his relationship with Dumbledore. Dumbledore might be more hesitant to trust Harry with the fight against Voldemort as his abandonment could indicate some instability or unworthiness in Harry's character in this fanfiction version.
A really interesting plot could be that James and Lily, alive, work with Dumbledore to create a new school of magic. This school would focus on teaching students how to defend against the Dark Arts in a more practical and hands - on way. Harry would be one of the first students there. They could also develop new magical artifacts that could help in the fight against evil. And the family could have adventures all over the world to find the ingredients and knowledge needed for these new creations.
One interesting plot idea could be that Lily and James, alive and well, decide to train Harry in magic from a very young age. This would make Harry an even more powerful wizard earlier on. They could also use their influence to reform the wizarding world, starting with Hogwarts. For example, they might introduce new courses that focus on both light and dark magic, teaching students how to defend against the dark arts in a more comprehensive way.
They could be shocked at first. But then they might start to enjoy it as they see the creativity of the fans.
In this fanfiction, Harry's upbringing by his parents would lead to a different Hogwarts journey. He may not be as eager to prove himself as he was in the original story because he already has a strong family support system. His role in the fight against Voldemort could also change. Instead of being the 'Chosen One' in a more solitary way, he could be part of a family - led resistance, with James and Lily having a major influence on the strategies and actions against the Dark Lord.
It would shock many fans at first. Since in the original series, they are a loving couple. But some fans might be interested in seeing a different take on their relationship. It could also inspire more fanfics exploring different relationship dynamics within the Potter universe.
A lot of fanfictions in this category might explore different timelines. For instance, some might have them survive the initial attack and then change the course of the wizarding war. They could use their combined magical powers to turn the tide early on. So, look for stories that focus on these aspects when you search for fanfictions about them being alive.