The differences between Goku and Vegeta stemmed from the environment they grew up in. Vegeta grew up in Planet Vegeta, a planet of the warrior race - The Saiyans, where one's social status and promise is determined by their strength from birth. His experiences growing up strengthened his belief that a warrior's promise is set in stone. Whereas Goku grew up on earth, where he was taught that a position or title is not something to be content with, that you can achieve greater things if you put your mind into it. And, similar to Vegeta, all of his experiences strengthened his belief with the difference being that Goku believed hard work is more important than social status.
The Saiyans being a prideful race made Vegeta extremely proud of himself. He enjoys the idea that his abilities are set in stone more than any other Saiyan since he is the prince, and being the prince means he has the potential to achieve something legendary and becoming the greatest in the universe. Because Vegeta defines himself by his social status, he displays a great deal of arrogance and puts down others for the sake of his ego. Whereas Goku is proud of the fact that he has limits to overcome. He enjoys the idea that his abilities are only limited by his own mind, and that there's no limit he cannot overcome, seeing as how he has no social status and is the most pure Saiyan, he would rather humble himself to have more opportunities in training than put down others.
When Goku and Vegeta clash so did their beliefs as fighters. Vegeta tried to put down Goku like he would other lower class warriors but Goku, thanks to his upbringing, was able to turn that on his head by humbling himself, saying he's actually grateful to the Saiyans for seeing him as nothing more than a lower class, because that allowed him to experience what he did on earth. By defeating Vegeta, Goku's beliefs outdid and shattered Vegeta's beliefs, with the events of the Namek saga damaging Vegeta's beliefs even more. (More insight on that on my Vegeta analysis http://www.kanzenshuu.com/forum/viewtop ... =7&t=31401 )
But to fully understand the difference between Goku and Vegeta's mentalities in their training one must examine the events of the Cell arc. Vegeta at this point was wrapped in his desire to be the best and outdo the one he hates. Despite being a genius like Goku is, his mindset greatly hinders his abilities. Vegeta desire to train hard was fuelled by his hatred whereas Goku's was fuelled by his pure desire to train. Because of this, Vegeta couldn't have hoped to attain what Goku did because to achieve the MSSJ form one must allow the MSSJ form to constitute their base form, which Vegeta couldn't fully do since he must eliminate the uneasiness and anger that stems from the SSJ form. Goku doesn't stress himself out and trains smart whereas Vegeta allows his pride to consume him and train more than necessary, which ends up backfiring at him.
When Vegeta went Majin he knew it was his only chance to beat Goku. At this point in Vegeta's life, it was no longer about proving that his beliefs as a warrior are better than Goku's, it was about regaining what pride he has left from the man that took it from him. Thanks to Goku, Vegeta was left broken and actually had to bond with his family. Something that Vegeta believes only weaklings do. Vegeta was forced to go against what he believed makes a strong man merely to have the will to live. But after reflecting on what he's become, he realized he's become a better man than before, he reached a level far beyond many Saiyans throughout history and although he didn't reclaim his birthright, he was lucky to have someone like Goku regain the pride of the Saiyans for him by defeating Frieza. And although it hurts him, it was Goku making him look beyond social statuses and Saiyan society that moulded Vegeta into a better warrior than he would've been otherwise. If he wasn't inspired by Goku to work on his senses and ki control he wouldn't have went as far as he did on Namek since Frieza's army/Saiyan society were fixated on their scouters and numbers to determine one's potential. If it wasn't for Vegeta's desire to become better than Goku he wouldn't have become as strong as he is now, and if it wasn't for Goku deciding to stay dead he wouldn't have taken the time to bond with his family and have something to fight for other than himself. Vegeta realized that, as much as it hurts him to admit it, Goku moulded him into a better man and Vegeta himself is thankful for it. Hence why he sacrificed his life for Goku as well as his family because in a way, Goku was Vegeta's most important friend.
Although Goku wanted to spare Vegeta to challenge himself against him, part of him wanted Vegeta to become good so he can have a fellow Saiyan to be friends with and spar against as well as being an ally that would help defend the earth, which was one of the reasons why Goku decided to stay dead. He knew that if he did Vegeta would be broken, but he also knew it was necessary so he would grow into a stronger ally and family man instead of leading a path that would end in his death and hell, which Vegeta himself admits in his speech during the Kid Buu fight. Vegeta came to terms with the fact that he cannot be better than Goku because he cannot match Goku in terms of upbringing and motivations. He realized at this point that Goku was his true inspiration, and although he was a better warrior than ever thanks to Goku, it was Goku's level of purity that inspired him, a level of purity that Vegeta admits that he can never reach. Which is the key difference between Goku and Vegeta and why Goku is "Number One".
