I have played Civ IV somewhat extensively but have only completed a few C3C games. From what I have been able to tell, C3C games tend to take much longer because there is a greater need for extensive micromanagement if you want to maximize your gains. I have found that the main cause of this problem has to do with the fact that there is no overflow in C3C like there is in Civ IV and the far increased number of cities combined with the corruption model and growth system of C3C also leads to more city micromanagement.
On the flip side, I find that wars in C3C are so much smoother than in Civ IV. No promotions to worry about and C3C encourages massive armies with its unit support system, something that Civ IV does not. I personally find using massive armies more fun over a small stack of highly individually specialized units, but YMMV.
Civ IV also has a memory leak problem. After playing for a while, the game simply gets bogged down and opening cities / giving orders to units tend to take several seconds. This can be fixed by alt-tabbing but it is very annoying to me. I have found no such problems with C3C.
The diplomacy system of Civ IV is much more mature. You can actually immediately tell why another civ does or does not like you and there are penalties for trading with worst enemies and so forth.
At the end of the day, I am probably partial to Civ IV because it's simply what I've played the most and have become best at. I also find that it's a more varied game - C3C tends to have fewer options and more general standard approaches to the map. In Civ IV, how the game unfolds is highly dependent on the map, so to be good you really have to adapt, which I find fun. Not saying that you don't in C3C, but it has been my experience that it is more important in Civ IV.
I do still play a round of C3C every now and then. But in the end, the city micromanagement gets to me.