As popular as this game was at the time, I don't think it got enough credit for its improvements over GTA 3. On the face of it, not much has changed, because the basic gameplay is almost identical, but everything else is so much better. Some the weaknesses of GTA 3 are actually strengths of Vice City, like atmosphere, mission design, and characterisation. GTA 3 had kind of a nondescript tone that doesn't really stand out when you go back to play it these days; but Vice City fully commits to a gorgeous and unforgettable 80s style. Missions in GTA 3 could get repetitive, vague, or frustrating; but Vice City's missions are mostly very creative and entertaining. While GTA 3 had a couple of memorable characters, none of them were very fleshed-out, and the protagonist was a mute with no personality. Vice City though? It has some of the best characterisation I've seen in a game. I'll never forget most of these characters, and they were all hilarious in their own special ways. As for the story itself, it mostly just borrows from a bunch of different films (Scarface in particular), but it's fairly unique to gaming and is very compelling.
Part of the reason the atmosphere in this game is so special is because of the soundtrack, which is hands-down the best soundtrack of any game, ever. One of the main reasons this game is important to me is because it introduced my young self to the joys of 80s Glam Rock, Pop, and New Wave, which helped to shape my musical taste, and is still shaping it today in the form of Synthwave. The music just makes the game, and it wouldn't be half as good without it, because it puts the cherry on top of the 80s theme. The radio stations are fabulous, and peppered with some of the funniest satirical commentary and fake advertisements you'll ever hear in a game. I'm also fond of VCPR, one of the talk show stations, which is so good I've kept it as an MP3 for years.
Apart from introducing a crouch ability, and cars feeling like they have a bit more weight to them, the shooting and driving mechanics are the same as GTA3, as mentioned above. I find the driving to be enjoyable in its simplicity and arcadey nature, but on-foot combat can feel bad these days. The main problem is that you have to stand still while shooting most weapons, which is awkward, breaks the flow of combat, and leaves you open to harm. Other times the controls get in the way are when you're piloting helicopters, which are hard to maneuver. There are two missions where you're forced to control an RC helicopter and an RC plane, respectively, and these are even more awkward to control. Those two missions are the only two I dislike.
It's one of my favourite games of all time though. The controls might have aged poorly, but everything else has aged like fine wine. It's just as gorgeous (artistically, thematically, musically...) just as funny and silly, and just as much pure fun as the day I first played it in Christmas 2002. Every time I replay it, it feels like a breath of fresh air from whatever bland, cookie-cutter nonsense the "AAA" industry has shat out in the moment. You will need to do some tweaking to get this running well on modern systems though, so check PCGamingWiki.