The two games are fully independent of one another -- you can play Vice City without any knowledge of the predecessor, though you'll miss many of the in-jokes if you do. The game engines are structurally similar; think of Vice as GTA3.75, not as a dramatic technological leap over GTA3.
Both games are masterpieces, frankly, and should be in your collection. I'd start with GTA3, particularly if you can find it preowned or otherwise cheaper than the full $50. Vice is a better game in many ways, but you'll appreciate Vice more if you're familiar with GTA3.
"When I feel depressed, I sit under a willow tree by a cool river, and imagine that I am strangling a duck." -- Kotaro Sarai