Was it simply that the density and/or kinetic energy got too low, because of expansion? Or was it something about the weak force itself (which I thought gelled around $10^{-10}$s?
For the CMB, photons decoupled because charged particles disappeared (recombined). But for neutrinos, their reactions involved electrons, positrons, protons and neutrinos, all of which were still abundant. Just too diluted, or what?
before one second, the density of the universe was high enough that interactions between them and the other constituents of the universe were common enough to put them all into thermal equilibrium, even though the interaction cross-sections themselves were tiny. After one second, the neutrino energies and the density of the universe had dropped enough that the neutrinos ceased interacting, dropped out of equilibrium, and went their own way. See Rob Jeffries' detailed analysis below about why the energies of the neutrinos themselves played a role in this process.
The general criterion for thermal equilibrium is that the neutrinos must couple to the radiation field (via charged particles) on timescales that are small compared with the timescale on which the temperature of the universe is changing.
Whether this condition is met depends on both the energy of the neutrinos and it depends on the density of the reacting particles too.
You can express all these factors in terms of temperature. The interaction cross-section goes as $T^2$, the number density of particles goes as $T^3$. The interaction timescale therefore goes as $T^{-5}$. The timescale for the universe to change temperature is the inverse of the Hubble parameter and scales as $T^{-2}$.
For thermal equilibrium, the interaction timescale must be shorter than the timescale for temperature change, but you can see that the ratio gets larger with time because it is proportional to $T^{-3}$.
The constant of proportionality is a temperature of about $10^{10}$ K, which is reached after about 1 second. After that, the neutrinos are not energetic enough and the universe not dense enough for them to interact frequently enough to stay in thermal equilibrium with baryonic matter.
