If you’re not intimately familiar with the different iterations of Civilization, this is a series which has done a fantastic job of carrying its core gameplay over with each successive game. However, there are still enough differences to make each game unique. It has been said: “If you play Civ 4 like you’re playing Civ 3, you are bound to lose.”
So with that said, I want to detail some of those differences.
-Government Selection Civ III’s Government selection system is completely different from it’s successors. Government options are unlocked through research. A new government can be implemented when its unlocked with a short period of anarchy to follow. Civ 4 replaced this with a civics grid. Civ 5 featured a social policies tree.
-Palace View The palace view is a small peripheral feature in which the citizens of a civilization would grant an upgrade based on performance. The upgrades were seemingly random and ultimately only served to alter a photo which was accessible from menus. This feature was available in Civ II as the throne room but was gone from the later games.
-Corruption Mechanic This is a big one. Corruption is a figure of wasted production and income from each city. Some of the things that effect corruption levels are government selection, distance from capitol, and buildings present in the city. (Courthouses reduce corruption.)
-Movement This was the last game where movement orders were given with the left click. Civ 4 and 5 both instituted a right click movement order system.
-No Unique Buildings Each civilization works with the same available building pool. Unique buildings weren’t featured until later games.
Here is a small list of other changes I found on Civ Fanatics
• Fewer great people (only Military and Scientific). And they are random. • There is a four turn minimum to learn any tech via research. • There are four distinct Ages. Civ 4’s ages function completely differently. • Fewer special resources. And no special building needed to use them. • Larger empires (100+ cities). • Come from behind wins are possible and fairly common.
This is the official webpage of Civilization III. As you can probably imagine, it’s dead now. However the downloads are still intact. Consequently there are a couple mods and maps that may be worth a look.
CivFanatics - http://www.civfanatics.com/ This is the central cultural hub for fans of the Civilization series. The forums are extensively broken down by game. If you check the Civ III section of those forums, there is a place dedicated to playing online. This is a great place to look for getting that elusive online multiplayer game going.
Edit 6/14/2013: Changed the Throne Room section to the Palace View section.