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.

-Colonies
Workers build colonies on resources to use them if they are not within your borders, for example tobacco that's nowhere near any of your cities.
Thanks to Megatronism for this info

-Map Size & Max Cities
The max cites possible to control is 256; more than most Civ games. The map size is also massive compared to later games.
Thanks to Greengoo for this info

-Early Ocean Exploration
You can choose to explore ocean maps from the very start of the game. Every time you end a turn with an early tech boat out at sea you risk having them be destroyed. This introduces a risk/reward tradeoff of trying to cross an ocean and establish a colony before anyone else get's there that is not present in other iterations of Civ.
Thanks to eggsome for this info

-Here are even more changes provided in the comments by user innadiated

• Civ3 has very unique border management. Nothing prevents opponents from entering your territory even if you ask them to leave.
  
• No strategic resource limit: Once you have iron you can build as many units needing it as you want.
  
• Roads earn you money, while in other Civs they cost you money.
  
• Has flood plains/marshes which will make nearby cities sick (and can even end up being your starting location!). Health was reused in Civ4 in addition to happiness, but flood plains and marshes were removed.
  
• You start with a settler and a worker, future games start with a settler and a warrior.
  Thanks to innadiated for this info

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.

Civ III: War Acadamy - http://www.civfanatics.com/civ3/strategy

This is a page from CivFanatics which goes into great detail about how to be successful in Civ III.

CIV III Section of the 2k Website -
https://civilization.com/civilization-3

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.
Edit 1/1/2018: Original homepage link was broken. Replaced it with the Civ III section of the 2k website.
Edit 8/8/2018:Added the bit about Colonies Megatronism mentioned in the comments. Thanks again for that!
Edit 9/11/2019: Added Greengoo's comments.
Edit 9/11/2020: Added eggsome's and innadiated's comments.
Edit 9/27/2022: Made sure the links still work.
Edit 11/06/2024: Made sure the links still work.