What sets Vercetti apart from his fellow protagonists is his drive, his ambition. He acts solely for his own personal gain, whether motivated by self preservation, expanding his criminal empire or to make a quick buck. He does not allow anyone to cause him to deviate or falter in his objective of seizing total control of Vice City's underworld, regardless of their standing as a powerful mob boss or Colombian drug kingpin.
I don't want to understate the effect GTAVC has on why Vercetti is so memorable. Although I don't agree that people prefer VC because it is colorful and vibrant, it definitely lends a LOT to Tommy's identity. The loud Hawaiian shirt. The pastel tone Soiree. The neon drenched Pole Position and Malibu clubs we purchase. They add to and shape the personality of a character who is essentially a dark Liberty City ex-con with an even darker past, in the same way you know you are "home" when you are rolling down Grove St on a BMX at sundown with CJ.
It also helps that GTAVC wants you to interact with it, wants to show you Vice City through Vercetti's eyes. Only GTAIV surpasses this, where Niko adapts to a whole different culture and as players we learned aside him with all that was new.
All the little things in GTAVC are geared towards the protagonist's personality - and that just makes it for me. When in the other games were we presented with a roaring blade and group of pedestrians happily chatting away nearby when we obtain a chainsaw for the first time? When else were we introduced to new vehicles with the protagonist actually encouraging us to steal a parked motorcycle? Who else of the protagonists would make scathing remarks immediately after a cutscene? It felt like you were in Vercetti's shoes.
I doubt Tommy would've been so memorable if he wasn't the first protagonist in the series to have a voice, though. I wonder what the effect this has on Tommy's popularity purely for nostalgic purposes?
It's worth remembering how short GTAVC was next to IV & SA in terms of main storyline. The game was released not long after III and had absolutely zero filler - especially when compared with the Badlands/Desert missions in SA and the "repetitive" go-here kill-him from IV that people often cite for disliking Niko and CJ. How much does that impact on Tommy's popularity that his story is short enough for him to retain his credibility?
Credit where credit is due to Ray Liotta - Vercetti wouldn't have had half as much snap if it wasn't for Liotta's powerhouse performance. I struggle to imagine anyone bringing Vercetti across as well as him.
TLDR? Vercetti took no sh*t, bowed to no one and was propelled to GTA legend by some terrific voice acting.