Well, one possibility is that Naruto would grow up with more focused and intense training. Jiraiya could pass on his unique jutsu more directly. Maybe Naruto would learn about the world outside of Konoha in a more in - depth way earlier on, traveling with Jiraiya and having adventures that shape his understanding of the ninja world and his place in it.
Well, one possibility is that Naruto would become an even stronger ninja faster. Jiraiya could train him more intensively from a younger age. He'd also be exposed to more of Jiraiya's wild and carefree ways, which might make him more confident and a bit of a prankster like Jiraiya. And Naruto would probably have less of the loneliness he felt in the original story, since Jiraiya would be a more present father figure.
In a fanfiction where Naruto gets adopted by Jiraiya, Naruto's training would be intense and unique. Jiraiya has a lot of unorthodox methods. He could introduce Naruto to some very special and secretive forms of chakra control. Naruto might also learn more about the secrets of the Sage Mode much earlier. With Jiraiya's guidance, Naruto could form stronger bonds with other powerful ninjas that Jiraiya knew. This would give Naruto more allies and a greater understanding of the political and power dynamics in the ninja world.
In such a fanfiction, Naruto might receive more focused training earlier. Jiraiya could pass on his unique ninjutsu skills more intensively, and Tsunade might teach Naruto about medical ninjutsu. Also, Naruto would likely grow up in a more stable family - like environment, with the support and love from both Jiraiya and Tsunade. This could lead to him being more confident and less of an outcast in the village.
Naruto would likely be more self - assured. Jiraiya's influence would make him more aware of his potential earlier. He might not have the same insecurities about being alone as in the original, since he has Jiraiya as a father figure.
His relationship with Sasuke would be different. Since Naruto would have a more stable upbringing, he might approach Sasuke with more understanding and less of the desperate need to bring him back. His friends in Konoha would also see him in a new light, as he is now under the care of two very respected shinobi. And his relationship with the village elders might be less strained, as Tsunade could act as a buffer.
If adopted by Jiraiya, Naruto would probably be more self - assured. Jiraiya would instill in him a sense of pride in his abilities. He'd also be more of a free spirit. Jiraiya's influence would mean Naruto would be more likely to take risks and explore new things without as much hesitation as in the original story. He might also be a bit more flirty, as Jiraiya has his own charm when it comes to the opposite sex.
Naruto's relationships would be different. He would have a father - figure in Jiraiya. Jiraiya could also introduce Naruto to new friends and allies. And Naruto would have more time to learn about his family and the secrets that Jiraiya might know about the Uzumaki clan. This could give Naruto a stronger sense of identity and purpose.
Jiraiya might comfort Naruto by sharing his own experiences. He could tell Naruto about the times he faced difficulties and how he overcame them. For example, he might talk about his training with the toads and how he grew stronger through perseverance.
Perhaps in this fanfiction, Jiraiya discovers that Naruto is going to be the cause of a major catastrophe in the future. He might have seen a prophecy or some sort of sign that convinces him Naruto has to die to prevent this. It could also be that Jiraiya has been misled by false information about Naruto, which makes him think that Naruto has become an evil force that needs to be eliminated.