Sorry, it took me a bit to finish reading Ultimate Campaign. All right, a comparison of the kingdom building rules in "Of Cities and Kings" from Rivers Run Red, the second issue of the Kingmaker Pathfinder Adventure Path, and the kingdom building rules from the Pathfinder RPG hardback Ultimate Campaign.
They are basically the same rules, slightly expanded. What is 10 pages in Rivers Run Read is 30 pages in Ultimate Campaign, plus some pages of optional rules.
Some of that is more options - like in terms of Leadership Roles, Rivers Run Red has 11 and Ultimate Campaign has 14 - and some of it is padding; for example the Councilor entry has the same mechanics but 6 lines of fluff verbiage rather than 1. So that additional page count is part more stuffs and part blabbering (though if you were confused by the sparsity of explanation in the Kingmaker version, that's not necessarily bad).
The "Improvement" phase gets renamed the "Edict" phase and other such, but in general you're getting the same system with a bit more of each of the lists-of-bits. There are some more buildings; you can do terrain improvements (waterways, bridges)... What it doesn't appear to have a bit more of is playtesting. Same system, most of the same holes. They did do bolt-on fixes to plug some of the most well known holes, like in the Income phase you can no longer sell magic items created by your city (no real in-game reason given, just to stop the "infinite monies exploit" popular among Kingmakers). They also took the Economy +1 and left only the Loyalty +1 benefit on graveyards, to avoid the Necropolis Gambit. So the various costs and DCs are left 90% the same, with some tweaks.
In general you are getting the exact same system in Ultimate Campaign (same grid layout page, same mechanics) with some additions and some cursory changes to prevent the most well known of the exploits in the previous system.