> What does Dark matter explain that MOND can't

MOND describes, it doesn't explain. By adding a simple function to Newton's law of universal gravitation, it does a very good job of describing the momentum-deficit in the bulk motion of large optical and radio sources in a variety of galaxies compared to the predictions from Newton's unmodified theory. That it does so, and with such a simple modification, is interesting. Milgrom's function in MOND is almost certainly the simplest post-Newtonian gravitational term extending Newton's theory consistent with observations. (There are other adaptations of Newton's theory to add post-Newtonian elements, for example to describe motions within the solar system. These are generally much more complicated than MOND [see [1] for a quick overview].)

In practically any post-Newtonian term expanding Newton's law of universal gravitation, one will discover a greater mismatch with observation as the relative speeds of material participating in Newtonian gravitation increase.
One has to "patch" with additional terms and counter-terms, resulting in something comparable to a Taylor Series. One generally experiences obvious failure with this approach as the relative speeds of the masses become comparable to tenths of the speed of light.

MOND doesn't describe the momentum-deficit in the motion of large clusters of galaxies, nor the momentum-anisotropies in the cosmic microwave background. It doesn't describe large mass-flows, or especially relativistic mass-flows as seen in Seyfert galaxies and a number of galaxy clusters. It doesn't describe fast binary objects. It doesn't accurately describe the fine details of the bulk sources that it describes very well. It also doesn't accurately describe our solar system; one has to "patch" MOND with additional terms, and it quickly becomes easier just to discard MOND as a contribution to solar system dynamics altogether. MOND also doesn't abolish the need to use General Relativity to describe the angle-brightness-redshift relationship of the reddest galaxies (and the supernovae within them) visible in the sky. MOND does not provide sufficient post-Newtonian correction for GPS and other orbiting satellites with very precise frequency generators on board, nor for the results of experiments in MESSENGER [3] and numerous other spacecraft.

Let's return to what MOND does do very well: describing the relative angular momentum of slow-moving inner and outer bulk bits of galaxies.

Attempts to adapt MOND so that it accurately describes relativistic galaxy/cluster components have generally resulted in adding a field to standard General Relativity [3], and it is mostly cultural history that keeps MOND from being properly promoted as a source contributing to the stress-energy tensor. That would put it on identical footing to dark matter in the bulk, focusing minds on the question: "what are the microscopic details of the beyond-the-Standard-Model-of-Particle-Physics contributions to the stress-energy tensor?".

> epistemologically far superior

What is Newton's explanation for his law of universal gravitation, which MOND slightly modifies in a very specific way? What is the modern explanation of gravitation, which can encode MONDian behaviour in various ways?

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[1] https://www.researchgate.net/publication/327607544_Post-Newt...

[2] https://pgda.gsfc.nasa.gov/products/66

[3] Chapter 7 of http://adsabs.harvard.edu/abs/2012LRR....15...10F (arxiv version: https://arxiv.org/abs/1112.3960)