Why does the Chandrasekhar limit affect white dwarfs differently? The Chandrasekhar limit is the maximum mass of a stable white dwarf star. Beyond this, a carbon-oxygen white dwarf will typically explode in a type 1a supernova, due to the nuclear reactions at those temperatures.

I've heard that oxygen-neon-magnesium white dwarfs, on the other hand, will not ignite. Rather, electron capture becomes energetically favorable and they become neutron stars. If I understand correctly, this also happens for carbon-oxygen white dwarfs in binaries, if the white dwarf has most of the mass in the system.

Why is this? Can anyone walk me through, step by step, what happens if a white dwarf (of both compositions) exceeds the Chandrasekhar limit, and how each differs?



Source: https://arxiv.org/abs/astro-ph/9701225
There are a variety of white dwarfs with various compositions, and analysing how they detonate in a supernova (or not) is an topic under investigation. A simple model, described in "How is the first detonation in Supernove type Ia triggered?" is of a helium shell initially igniting and that setting off the carbon in the core.

In this type of Type 1a supernova there is no neutron star formed, as the star is completely destroyed. The mass at which a white dwarf will undergo a type 1a supernova is just below the chandrasekhar limit, so neutronisation won't occur.

However in some white dwarfs with an atypical composition (as you note a Mg-Ne-O white dwarf) it is possible for the star to avoid detonation, and reach the chandrasekhar limit, and so for electron capture to occur, and a neutron star to form. It is worth noting that there is not definite observation of a white dwarf collapsing to a neutron star (where as there a lots of observations of type 1a supernovae) however these "accretion induced collapse" scenarios may explain some magnetars and short gamma ray burst 

Progenitors of the Accretion-Induced
Collapse

So the two scenarios are, 


Accretion occurs onto a Carbon-oxygen WD. The pressure and temperature in the core of a white dwarf increases until thermonuclear reactions begin (at about 1.38 solar masses). Since the white dwarf is degenerate it can't expand to reduce the rate of thermonuclear reactions, and the entire star detonates and is destroyed.
Accretion occurs onto a ONeMg WD. The star reaches 1.44 solar masses, electron degeneracy is no longer sufficient to prevent collapse. Electron capture occurs and the star collapses to neutron star.