Rhapsody is Mac OS X before they thought of Carbon...
Carbon is a reimplementation of the Classic Mac OS APIs on top of a modern OS. Carbon was based on QuickTime for Windows.
When Apple was porting QuickTime to Windows they decided it was simpler to port most of the Mac OS Classic API to Windows. Doing this required a cleaner reimplementation of most of the Classic Mac OS, and the underpinnings of Windows at the time were vastly more modern than Classic Mac OS. Using QuickTime, you can port Mac apps to Windows without a complete rewrite. iTunes for Windows today still runs atop QuickTime for Windows. Carbon for Windows if you like...
With Rhapsody, developers were required to rewrite their Apps in Cocoa to gain any benefits of the new OS. Existing apps could run in a virtual machine codenamed "Blue Box" - which later became "Classic" in the released Mac OS X. Asking developers to rewrite obviously proved unpopular.
Carbon was the solution, which enabled developers to update their code with considerably less effort than a complete rewrite, and they would gain the benefits of the new OS. Carbon is still used today by many high profile apps (Final Cut Pro, iTunes, Photoshop, InDesign, Illustrator, Microsoft Office... etc)