supernova - Why does the Chandrasekhar limit affect white dwarfs differently?

Whether a white dwarf responds to the accretion of material by exploding or collapsing depends on the competition between energy being released in fusion reactions and energy being locked away by endothermic electron capture (neutronisation) reactions.

It is thought that most white dwarfs of moderate mass have a C/O composition. They will need to accrete a lot of mass to get to a density (at about $4times 10^{13}$ kg/m$^3$, reached at $1.38M_{odot}$ in a non-rotating WD) where neutronisation becomes energetically feasible. It is possible, that before this happens, that fusion reactions are ignited (due to high density, rather than temperature). The threshold density for ignition is lower for nuclei with lower atomic number (He < C < O) and the ignition threshold densities for He and C are probably lower than the neutronisation threshold for C.

In a C/O WD that has accreted a lot of matter, ignition could take place in C at the core, or it could be triggered in He (at even lower densities) at the base of a deep accreted shell of material. The electron degenerate matter has a pressure that is independent of temperature, leading to runaway fusion and the complete destruction of the star.

O/Ne/Mg WDs are made as the final stages of more massive stars ($8-10M_{odot}$) and are born as remnants with much higher mass $>1.2M_{odot}$ than typical C/O WDs. More massive WDs are smaller, with higher density. The neutronisation thresholds for O, Ne, Mg are only $1.9times10^{13}$, $6times 10^{12}$ and $3times 10^{12}$ kg/m$^3$ respectively (all lower than for C). This means that a O/Ne/Mg WD may have to accrete very little mass to reach this central density, begin neutronisation, which leads to collapse. In addition if such densities are insufficient to trigger C burning in a C/O WD, then they certainly won't be high enough to trigger burning in O/Ne/Mg because of stronger coulomb repulsion. Further, if little mass is accreted, then there won't be a deep envelope of accreted material in which to ignite burning off-centre.

For all these reasons, O/Ne/Mg WDs may be more likely to collapse than explode (the collapse would cause a type of core-collapse supernova though).

EDIT: Actually looking at the paper you reference (which is a bit dated), although some of the numbers have changed slightly, the semi-quantitative argument I give above is exactly how it is explained there. So I'm not sure whether my answer helps you.