The main protagonist of the Shonen-Anime, Dragon Ball, is Goku, the monkey-tailed hero born from the war-loving Saiyan race who was sent out from the Planet Saldana as an infant in order to conquer Earth. Raised by Grandpa Gohan without the influences of his bloodthirsty relatives, Goku grows up to become Earth's powerful protector, and it is many years before he meets another Saiyan and learns of his heritage - and his true name, Kakarot.
Vegeta makes his first appearance in Dragon Ball Z when he teams up with Nappa to find the legendary Dragon Balls in the hopes of wishing for immortality. It does not take them long to locate their fellow Saiyan living on Earth, although Goku's loyalties are aligned to the Z-Worriers as they defend their home planet. Vegeta was already aware of the Saiyan's presence on the planet and, like Raditz, refused to acknowledge Goku's foreign name, only ever referring to the Saiyan hero as Kakarot.
Bully Tactics
For all the DC fans out there, here's a simple analogy - consider Superman, who is commonly known as Clark Kent to Earthlings, but Kryptonians such as General Zod make use of his birth name, Kal-El. It is much the same with Goku, who is only identified as Kakarot by his fellow Saiyans. Insisting on calling Goku by his Saiyan name was initially done as a taunt, to remind the hero of his devastating ancestry.
It was quite obvious to Vegeta that Goku was less-than-pleased about being part of the destructive Saiyan race, and he got a kick out of constantly reminding Goku of where he comes from. Saiyans were, after all, raised with the sole intent of traveling across the Universe and conquering planets along the way, to be sold on the galactic market. As a peace-loving humanitarian, Goku could not be more opposed to the Saiyan ideals and wished not to be associated with them.
Saiyan Pride
Being a Saiyan Prince and one of the last few surviving pure bloods of his race, Vegeta is honored by his heritage, and pride is more important to him than mind games. It would be impertinent to call his kin by any Earth-given name, leaving his Saiyan ancestry unacknowledged. Using the name Kakarot has more to do with paying homage to his race than showing Goku respect.
Additionally, because only a handful of pure-blooded Saiyans remain in existence, to not use the name Kakarot would be almost sinful. Raditz, Vegeta, and Nappa referred to Goku by his birth name because, as far as they are concerned, the hero is still a Saiyan regardless of where he grew up.
Baby-possessed Vegeta
Baby is a parasitic Machine Mutant created by the Tuffles who is able to possess a host, taking over its body through cellular regeneration. Baby is generally able to go about undetected, often hiding in plain sight as he spreads his eggs in an attempt to gain enough forces to revive the Tuffle race. The Mutant manages to overpower Vegeta and take control of his body, creating another spy within the Z-Warrior force. However, Baby is quickly discovered when he makes the mistake of calling Goku by his Earth name, causing immediate suspicion and dispelling the deceptive disguise. So Vegeta's stubborn use of the name Kakarot does pay off in the end.
The only other time that "Goku" escapes from Vegeta's lips when conversing with someone unfamiliar with the Saiyan name, and even then it is done indirectly -"known on Earth as Son Goku" is the closest Vegeta ever got. Ironically, however, in Dragon Ball Super: Broly, Goku himself requests that Broly calls him Kakarot, finally professing his connection to the Saiyan race. As one of four remaining pureblood Saiyans, (five if you consider the possibility that Vegeta's brother, Tarble, may still be alive), Goku appears to disregard his disdain for the Saiyan race and instead find kinship within his lineage.
Despite the time that they spent together throughout Dragon Ball, as enemies or friends, Vegeta has never wavered from using the name Kakarot. This, however, does not seem to bother Goku because it is a reminder of the bond that the two Saiyans share, having fought alongside each other for so many years. Goku and Vegeta's kids have subsequently grown up together and, for all intents and purposes, they are all one big family, 'brothers' with their own little pet names.