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Difficult one to answer, depends on your personal preferences to some degree, however, in an effort to give some overview:

AmigaOS is a reimplementation of Commodore's OS 3.1 operating system, built with access to the original source code which in many areas remains unchanged, however using a new kernel (ExecSG) built by Hyperion sometime after 2001. It runs on AmigaOne, Sam, and Genesi PowerPC systems (SE, XE, micro, Sam440-ep/flex, Sam460ex/cr, X1000, Pegasos II). There is a 'classic' version that runs on Commodore Amigas equipped with a PPC expansion.

MorphOS is a reimplementation of Commodore's OS 3.1 operating system, built by reverse engineering the original components and reimplementing them, using a new kernel (Quark) built by the MorphOS team (mainly Ralph Schmidt in the early days afaik) sometime after 1998. It runs on Sam, Genesi and Apple PowerPC hardware (Sam460-ex/cr, Pegasos I, Pegasos II, nearly all Apple G4 desktops/laptops/minis/all-in-ones, some PowerMac G5s).

Both are currently limited to 32-bit mode when running on 64-bit processors. Both are currently limited to running single-threaded on one core, even on multi-threaded/multicore processors.

AmigaOS4.x was built by the people who ported Freespace and Shogo from linux to AmigaOS, with much help from Olaf Barthell and many others.

MorphOS was built by the programmers behind the finest Amiga expansions (Blizzard, Cyberstorm), RTG subsystem (Cybergraphics), and kernel extensions (PowerUP), amongst many others.

AmigaOS has support for some more recent graphics cards than MorphOS, and has some features (eg Paula audio chip wrapper/emulator) that MorphOS does not.
AmigaOS runs on more 'new' hardware than MorphOS (custom hardware, v expensive sadly), MorphOS supports a vast range of commonly available Apple (PowerPC only) hardware which is easily obtainable, replaceable, and affordable.

When running on the same hardware MorphOS outperforms AmigaOS in most (all?) areas, however AmigaOS doubtless has some features that MorphOS lacks.

Sorry if that seems a bit one-sided. :)

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They are emulation packages which can be run on any system running Windows, or whatever the system requirements are. Either way, they essentially run as a program on Windows/whatever, cleverly hiding the use of windows apis and drivers to access system resources.

MorphOS (And AmigaOS, and AROS (native)) are operating systems which directly access the hardware using their own kernel/drivers, not relying on a host OS underneath.

AmigaOS and MorphOS, both of which run on PPC processors only, include in-built 68k 'emulation' to allow them to run binaries built for the Amiga series of computers which used a 68K processor. In each case this 'emulation' translates the 68k code into ppc code as it is run - translating the code into native ppc instructions rather than actually emulation a 68K processor (if I understand rightly).
The translation is invisible to the user, 68k binaries run just like ppc binaries.

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Individual preference may vary. There are some programs on AmigaOS that seem interesting which aren't available on MorphOS, and vice-versa.

Games, especially opengl ones, will probably be quite a bit faster in general on MorphOS due to AmigaOS rtg limitations and MorphOS having much faster graphics drivers (comparing performance on the same hardware).

Development cycle/updates are a lot more frequent and polished on MorphOS. User feedback and bug reports are acted upon more quickly in general.

Users and app developers of each OS are friendly and grateful for feedback.

MorphOS devs have never (publicly) wished death upon a large proportion of their users! :D

In summary, it's a topic that could easily descend into acrimony. I'm sure asking the same question at Amigans.net would lead to a similarly one-sided perspective. (if the thread didn't get locked first. :P)