Ataxerxes has a good list. A few others things (with a bit of a military focus here):
- Roads give you the commerce bonus in Civ3, so build them everywhere that's being worked.
- Railroads' infinite movement means that you have to be a little bit more careful with border defences once your rivals have them - if your cities are too sparsely defended, they could potentially strike quite deep into your territory in one turn with Cavalry.
- In general, the offensive is a bit more powerful in Civ3 than in Civ4. In Civ4, the defender always gets the ideal counter unit from their stack to fight (ex. attack with a Knight, the Pikeman fights). In Civ3, there's no bonuses against certain types of units, resulting in the offence doing a bit better. The big exception is the Great War era, where defensive Infantry dominate.
- Artillery works differently in Civ4 (and I think, better). They bombard from a distance, and don't suicidally charge all by themselves. They can also be captured if inadequately defended.
- Similarly, while artillery can be a big help, especially at higher difficulties, you don't need artillery to take cities like you often do in Civ4. Throwing in a few extra Swordsmen or Knights if the city is on a hill will also suffice for taking it - you just might lose a few of them. (This becomes less true once your enemies reach the Industrial Age and size 13+ cities)
- Bombers are a lot more versatile in Civ3. They can bombard buildings in cities, destroying them as well as the population. More importantly, they can also bombard tile improvements, including roads. This is especially potent in Conquests/Complete (and maybe PTW), where craters are created once all improvements are destroyed (these reduce the tile's production to below its base value). Capital ships and artillery are similarly able to bombard tile improvements, but of course at lesser range.
- Cavalry (and later, tanks) can withdraw on the defensive as well as the offensive. This can make it a bit more difficult to fight a cavalry-fielding foe.
I think that covers the primary wartime differences. Hopefully that will help win some wars!