Civ 3 / Civ 4:
Fun: For me Civ 3 is much more fun to play than Civ 4. From time to time I test new Civ 4 mods, especially epic mods to see, what´s now the "state of fun" in Civ 4. With the last new Civ 4 mod I tried, it was the first time I had the "one more turn feeling" in Civ 4 - but only for a shorter time, than it became boring again. During the "fun part" In that game I was in war from the first turns of gameplay until I eliminated all enemies on my continent. But also this was not the same kind of fun I have in Civ 3.
Graphics: A lot of people say, the graphics in Civ 4 would be better than in Civ 3. Some people, among them me, are saying, the graphics of Civ 4 are very ugly compared to the graphics of Civ 3.
May be it´s necessary to have a closer look what is meant here. Civ 4 has animated waves at the beaches and rivers, when a terrain improvement is used by a citizen, that improvement is animated (for example smoke coming out of the chimney of a mine) and when walking through forests, birds are escaping from you - but: These animations can only be seen if you have a certain factor of zooming-in, and this degree of zooming-in is for a game, that focuses on a strategic scale of gameplay, next to useless.
On the other hand most of the Civ 4 units in my eyes are looking ugly and if they are zoomed in (a feature that doesn´t exist in Civ 3 gameplay) they are looking even more ugly. Why? Because they must use low polygone graphics, due to the 3d-engine used in Civ 4. Wouldn´t you look somewhat "ill", too, if every second atom in your body would be missing?
The modded terrain graphics in Civ 4 themselves (without the animations) in my eyes are not very impressing, compared to the modded terrain graphics for Civ 3. Especially the mountains in Blue Marble are not looking very convincing to me.
Combat: So this belongs to the section "fun", a lot of Civ 4 supporters have a tendency to focus out here a special advantage of Civ 4. In Civ 4 there are some additional flags for units in combat that are added to the stone-paper-scissors-principle. These flags exist since Civ 2 (pikemen flag). They were not used in Civ 3. Why? Civ 3 has a special system allowing to set special attack targets for every land- and seaunit, called "stealth attack". In C3C and in most mods this feature was never used well, as most Civ 3 modders didn´t recognize, that this flag is only working, if a stack contains at least two allowed targets for a "stealth attack". The development in stealth attack, if there is only one allowed target in a stack, was cancelled due to the development of Civ 4.
A propos stacks: The suicide catapults in Civ 4 in my eyes are an "unworthy" solution to that problem for programmers, who have access to the source code. I think the approach in Civ 3 to that problem (for civers who consider monsterstacks a problem) was much more subtile: A special kind of bombardement, that has a a chance of hitting each unit in a square and diminishing the value of defense for those hitten units by 50 % for that round of combat (if a unit was hit by that kind of bombardement no further reduction in the value of defense was possible in that turn).
The programmers in C3C named that function "charm attack" (please don´t ask me why). It is in Civ 3 and can be brought to function with a special editor, but the images and sounds for such an bombardement are not there (or were not found yet). The development on charm attack was stopped due to the development of Civ 4.
Firaxis and 2K did´nt give us the source code of Civ 3 -not even to repair some selfmade glitches by Firaxis like the submarine bug- as it seems they are well aware, that Civ 3, especially if all these options are in full function, is a much better game than Civ 4 (and Civ 5).