From my experience, the animations in CIV 4 were actually very detrimental to game play. Say players France and Germany are at war, and the person playing Egypt is happily on the other side of the world knowing nothing about this war.. Unfortunately, because the Germany player had animations on, the game would seemingly randomly pause the turn timer countdown for every player in the game while it showed the animations to the German and French player. Poor Egypt would have to put up with constant delays while the animations did their stuff in a battle that he had no awareness of.
This is why in the MP community, we basically screamed at everyone to turn the damn animations off while they were playing MP.

In this case, they just didn't give you the option to get screamed at.