In case a dry restatement of the chronology helps to resolve the confusion:
NeXT Computer was acquired by Apple in 1997. At the time NeXT's most successful product was WebObjects, the Objective-C/Java server product; the NextStep OS had turned into OpenStep in partnership with Sun but the latter had drifted away after deciding to push Java as the solution for everything.
Rhapsody was both announced and demoed by Apple in 1997, one month before it completed acquisition of NeXT; it was subsequently made available to the public as two Developer Previews, the first in 1997 and the second in 1998. It provides a wonky emulation of the MacOS 8 look and feel atop the NeXTStep stack of the Mach kernel plus a BSD subsystem, with two 'boxes' for applications — the yellow box provides a version of the OpenStep libraries, for fully-native apps, and the blue box provides MacOS 8 in a box. Literally in a box; all MacOS 8 applications ran within a single common window, much like the experience you usually get running a virtual machine. A Java Virtual Machine is offered, but as with the two boxes that's once again a separate thing, in its own little API world.
From there Apple moved to OS X Server 1.0 in 1999. It still looks like MacOS 8, though it's still a reimplementation so the widgets don't always act identically, with NeXTStep-esque Display Postscript as the graphics layer, the NextStep Workspace Manager as the desktop, and classic Mac applications constrained within their own little desktop-in-a-window.
Darwin was also born in 1999, then being the kernel plus the BSD subsystem plus other parts that Apple publicised as producing a complete operating system with Apple's driver model. Over time it has sometimes tracked core OS X developments — e.g. switching to CUPS, gaining libdispatch — but it'd be out of scope to discuss more modern developments too thoroughly; see old release notes for evidence of its scope during the relevant period, and note that not everything Apple provided in its compiled versions of Darwin was provided as open source. So the popular version of events — that Darwin is the open source parts of OS X — also isn't completely accurate.
OS X 10.0, the first consumer-targeted release but still looking and acting a lot like a beta, didn't arrive until 2001. Significant changes:
- the Aqua user interface, in its lickable phase, has arrived;
- as has most of the rest of early OS X: Aqua implementations of the Finder, the Dock, and the menu bar;
- classic applications are no longer constrained to a box;
- Carbon has been added as a native framework (and, similarly, is available for the Classic OS) to provide the migration path necessary to retain the support of Adobe, Microsoft et al;
- as per the history with Sun, the Java bridge is present, providing Java with the same access to Cocoa as Objective-C; and
- various remnants of NextStep survive that will later disappear: the pre-CoreText font rendering system being probably the most prominent.
So to directly answer your questions:
- Darwin underpins macOS/OS X; but
- public OS X releases predate Darwin releases by two years; and
- there is a visible technological evolution in OS X during which direct NextStep components are slowly replaced (the Workspace Manager's mutation into the Aqua desktop via MacOS 8 controls being an obvious example), though none of those components are provided in Darwin.
So I think it's most accurate to say that OS X is the descendant of NextStep, albeit quite far removed by the time — four years after NeXT's acquisition — that it finally became a consumer product, and Darwin is a spin-off of OS X.
(and if I've been inconsistent in my capitalisation of NeXT/Next, it's because the company was famously indecisive)