I play 3 more personally right now. But I find 3 and 4 are about equally enjoyable. They play very different as to economic structure and empire management. But each have their advantages IMO. I like how wonders have been nerfed in 4. (Most only effect the city they are built in and nothing outside of it. Great library allows 2 free scientists instead of the usual free tchs from previous versions as an example.) I also like the new style of diplomacy added in, diplomacy in general is only slightly better than 3's. TO me 3 still holds some things over 4's system, due to the nature of the AI being more aggressive. (And no, aggressive AI is not near the same on 4) Eh, I will do this an easier way.
I would say the positives in Civ 4 are:
-Visible Diplomacy Model (You can see why someone likes or dislikes you)
-More trade options (Although mostly automatic)
-Variety of Great People with very different roles
-Wonders are not as powerful or game altering as before but still very effective.
-Bonus resources now benefit more if you build specific improvements on them. As well, resources now MUST have the specific impovement to gain the advantage. (Iron most not only have a road but also a mine before you can build swords.)
-Mod Capabilities are limitless.
-Better system using Happiness
and Health to influence population growth than a standard hardcoded number.
-Better multiplayer interface.
Where Civ 3 prevails for me:
-Better Graphics. (4 is extremely cartoony - although some units look pretty cool, especially scaling them down and using SeZereth's formation mod.)
-More vicious AI diplomacy if wanted.
-Spontaneous Great people
-Much faster and easier modding although limited.
-Much more interesting map generator.
-Better Artllery Combat. (Note: I have not played beyond Civ 4 vanilla and I am aware that BTS has made some changes to artillery and such so this may be an outdated response even according to my opinion. I plan on buying BTS as soon as finances allow but am not able to atm.

)
-There are seas, not just coast and ocean. (Seems small, but I
love the aspect seas bring in)
-Air and Navy combat.
-GREAT advisor screens. Except cultural advisor is about worthless for me. (Although I like how in 4 it shows everyones relations to everyone as default for the foreign advisor screen, and that there is room for everyone.)
-You can cross through people's territory and "dare them to kick you out" without initiating war. SOmetimes it is scary to ask someone to leave your territory.(In 4 stepping in someones territory triggers war automatically)
Civ 4 BTS may also include the original Civ 1 and Civ 2 spaceship model. Can someone verify this for me please?