For the "regular" game, Civ4 wins hands down. Civ3 is just the same rehash every time: settle land like crazy, conquer first neighbor with iron/horse based units, build armies and artillery and finish off everyone else. At higher levels it's all about tedious MM, building tons of crappy, tiny cities. No wonders, no infrastructure, no culture. Bleh.
Civ4 offers multiple ways to win, reduces MM and offers all sorts of strategic possibilties without the need for crutches like armies and artillery. My first win on Civ4 was by spaceship whereas the Civ3 end game is so boring, I never could be bothered to finish a space win. It was just so much easier to wipe out everyone. In fact, even wiping everyone out gets overly repetious and dull so my last several games of C3C I just stopped playing when it was obvious I was going to win anyway.
That being said, there are a lot of really fun mods of there for C3C and even the scenarios that come with C3C are pretty good. Civ4 mods are still experiencing their growing pains. Over time this will surely change and given that Civ4 is far more moddable, I suspect it will greatly surpass Civ3, especially in the area of scenario events, which were inexplicably left out of Civ3.
So for the time being, there's still some life left in old Civ3, but the game is clearly on life support.