Given Goku's long history throughout the Dragon Ball franchise of being… less than intelligent, it's almost surprising he can keep up with having two names. Perhaps it helps that Vegeta is the only one who ever calls him Kakarot. No, Kakarot is not the Saiyan word for "clown," though that would be an equally believable reason for his insistence on using the name. Still, there must be a reason Vegeta has refused to ever use his fellow Saiyan's Earth-given name -- other than stubbornness, anyway.
Goku and Vegeta first met on the battlefield as adversaries in Dragon Ball Z. Nappa had just massacred most of the Z Warriors when Goku finally arrived and beat the burly Saiyan so badly that Vegeta decided it'd be better to just kill him. During this time, Vegeta was openly evil and quite proud about it, too. Most likely learning of Goku's existence from Raditz, Vegeta may have taken to calling him Kakarot as a taunt to remind him he was of the same race as the monsters who'd come to Earth and killed his friends.
In the rearview mirror, it's strange that Vegeta so gleefully killed Nappa considering all the time since he'd been burdened by being one of the last two remaining Saiyans. Vegeta's pride has always defined him; nothing and no one is more important to him than his Saiyan heritage. Goku, on the other hand, has had no interest in his Saiyan ancestry after meeting his jerk of a brother and then the evil Nappa and Vegeta soon after. So, because Goku has opted not to embrace his identity as Kakarot of the Saiyan race, Vegeta has chosen to embrace it for him.
Perhaps unsurprisingly, the reason Vegeta calls him Kakarot has less to do with Goku than it does with the Prince of Saiyans himself. Saiyan civilization has been entirely lost since Frieza blew up their entire planet, and now the names Vegeta and Kakarot are the only remaining evidence that their culture ever existed in the first place. While Goku doesn't grieve over his lost race, Vegeta does, so referring to his only fellow survivor by his given Saiyan name is a way for the Prince to keep their people alive in spirit.
As their relationship has evolved in Dragon Ball Super, Kakarot has become closer to a sign of respect. After all, if Vegeta considered him an unworthy Saiyan, he wouldn't do him the honor of referring to him by that name. All these years later, Vegeta's attitude has even rubbed off on Goku. At the end of the film Dragon Ball Super: Broly, Goku specifically requested that Broly refer to him as Kakarot. A minor moment, but it's the first sign of interest Goku has ever shown in his heritage by seeking out a sense of Saiyan camaraderie with the only other living (canon) full-blooded Saiyan.
As much as Dragon Ball fans would love to believe Kakarot is a term of endearment from one best buddy to another, the idea that Vegeta is trying to keep his culture alive is far more profound. As with most things in Vegeta's life, using given Saiyan names is a way for him to express his pride. Names might have even been of great cultural importance, considering he had wanted to give his daughter a traditional Saiyan name and was visibly affronted when Bulma overruled him. Perhaps Bulma was just trying to spare her daughter from a life where she had a vegetable for a name.
