There's a single feature that makes GTAIII vastly superior to the rest of 'em:
Story IMMERSION.
Is not the story itself, or the soundtrack, is not the protagonist or the characters, nor the philosophy at hand. It's the quality in which all these factors melt together.
All 3 games have a great immersion. But III is the perfect mafia and crime game, and the city in which it's placed simply works better, as small as it is, as gray as it is, as echoey as it is and as dull the classical music can sound. All of those things make it pretty metal to enjoy it all. And heck even the mute protagonist works better this way, which was a very bold decision to adapt the game to this after they were left out without a voice actor.
I've been playing these 3 since they came out and never paid much attention to GTA V, but when I got bored of playing Vice City and San Andreas... there was always something that made it so fast and easy for me to jump again into III's world and advance the story. Maybe it's the lack of boring levels (like those with the RC toys in the other two games), or the lack of open-world activities, which in itself makes it so consistent and easy to focus. But you just can't get bored of it.