Nathiri gave a good overview, so I'll only touch on what I know enough about to discuss in more detail - maps. The absolute limit for maps is 65,536 tiles (traditionally 362x362, divided by two due to being isometric, although you can have more than 362 in each dimension). However, the practical limit is enough land to accomodate 512 cities, which is the maximum. There have been some attempts to circumvent that, but I'm not confident in saying that any of them have been fully successful. Still, that typically requires around 250x250 = 30,000-ish tiles, which is generally both more tiles and more cities than Civ4, which would run into memory problems before that point. I'm less familiar with the engine limitations of Civ 5, but 5 generally favors taller empires more versus the wider empires of III, and I don't recall seeing any Civ5 games with maps larger than the larger Civ3 ones.
From a playability standpoint, the map limitation is not RAM like in Civ4, but CPU speed, as a larger map and more AI civs slows down computation. Thus, on large maps with lots of AI civs, modders typically do things such as restricting which buildings enable water/air-based trade beyond the default, as trade calculations are among the largest CPU cycle culprits. Generally however, Civ3 is the most amenable to large maps with many cities of III -> V.
I'm not aware of any equivalent to Civ IV corporations, although it is possible to have buildings that generate set amounts of gold per turn, like in some Civ4 mods. This can be done simply by setting the maintenance cost to be negative. Adding new concepts in general is not technically possible (as it is in Civ4/5 with Python/Lua/SDK modding), but a number of mods have create semi-new features by using existing ones in creative, different ways. Civinator's micro-zones, for example, are a creative approach that was used in the Storm Over Europe mod to cause the AI to create a mix of units in roughly the correct proportions for the time period and strategy, despite it not actually being possible to modify the AI. They use strategic resource and strategic placement of roads/rails (and terrain) to enable certain cities to specialize in armor, some in aircraft, some in infantry, etc., resulting in about the right mix overall. I'm sure his initial post explains it better, but the gist is from the starting point of "the AI will produce what it calculates is the best unit available" he was able to achieve "the AI will create a period-and-strategy-appropriate mix of units", via creative use of the available mechanics.
On the whole, I find Civ3 has some of the most polished mods for Civ at large, particularly for historic time periods (where it also has a great variety of mods). I attribute this to having a fairly powerful set of tools that is fairly easy to use, allowing those whose expertise is in areas such as history to be able to create detailed, in-depth mods, without having to be programmers themselves.
Civ4 (and probably Civ5 too, although I've read it isn't as flexible with modding as IV) does allow for more powerful modding, including completely new concepts, and some mods have made good use of this. However, to really take advantage of that power requires being a programmer, and if you aren't a programmer, what you're left with is less powerful than what Civ3 offers. I think this niche has served Civ3 well over the years, as it helped it gain a sizeable library of mods and scenarios by allowing it to draw on a larger pool of talent, and a fair amount of that talent has stuck around for a long time. Complete overhauls like Fall from Heaven II aren't possible, but quite a bit of really-quite-good modding is.
