Carbon vs. Cocoa
Actually, both Carbon and Cocoa apps run in OS X. Cocoa is purely for X, and apps written in it can take advantage of certain features built into the OS (like spell check, services, font smoothing, etc). Carbon is used mostly for older apps that are being ported to OS X, like all the MS and Adobe software. While it's much more feasible for software developers to carbonize older software instead of totally starting from scratch, cocoa apps tend to be more responsive, etc *because* they've been written from the ground up for OS X.