There used to be a question (and answer) here that mentioned why isotopes of cadmium, xenon, hafnium, etc. that have larger cross-sections than boron-10 weren't as good at being neutron absorbers in nuclear reactors, waste sites etc. .....(besides cost)....
I think the answer(s) had to do with what the resulting isotope was; if the product isotope could also absorb neutrons; or if the daughter isotope released neutrons or dangerous fission products itself...
The issue is the secondary radiation emitted after the neutron is absorbed.
Most medium-mass nuclei have a "neutron separation energy" of about 8 MeV. When a nucleus absorbs a neutron at rest (which is a good description of a milli-eV thermal neutron in a mega-eV interaction), the separation energy of the compound daughter nucleus must be emitted, usually as a cascade of several high-energy gamma rays. Cadmium is especially nasty. If you put a cadmium sheet with any macroscopic thickness in a neutron beam, there won't be any neutrons on the downstream side of it, but there will be a boatload of hard gamma rays. This gamma radiation is prompt and must be considered before you worry about whether the daughter nucleus is stable against beta-decay or whether it will contribute to "activation" of your neutron absorber.
Smaller compound nuclei are also more likely to decay by strong particle emission instead of photon emission. For neutron capture on boron the primary interaction is
$$ \require{mhchem} \ce{ ^{10}B + n \to {}^7Li^{\star} + \alpha } $$
with generally a soft, half-MeV photon from an excitation in the new lithium-7 nucleus, but a good chance (~15%) of no photon at all. The strongly-interacting lithium-7 generally does not escape the solid matrix which holds your boron powder; the alpha doesn't escape until long after it has cooled to room temperature and started to act like helium.
The gold-standard absorber is lithium-6, which has a large cross section for
$$ \ce{ ^6Li + n \to {}^3H + \alpha} $$
in the presence of slow neutrons. Since neither the alpha nor the triton have any low-energy, photon-emitting excited states, lithium-6 basically turns a thermal neutron beam of any practical intensity into ... nothing. No electromagnetic radiation is emitted, and the gas atoms thermalize in whatever solid contains your lithium and then generally diffuse away into the air before accumulating in any chemically significant quantity.
The requirements for a neutron absorber and neutron shielding is different. The purpose of the latter is to protect people (or electronics) from radiation, hence the secondary products detailed in @rob's answer are a concern.
For shielding, borated paraffin blocks are common. Fast neutrons scatter of the plentiful hydrogen atoms, sharing their energy 50/50 (on average, of course) because protons and neutrons have similar mass.
The scattered proton is charged and quickly loses energy via ionization, while the neutrons are absorbed in the boron-10.
If you are looking for an neutron absorber for shielding calculations, it is almost always going to come down to cost vs. effectiveness. You can have materials that absorb twice as many neutrons, but they might cost 10 times as much. Some of the things you have to consider are:
In reactor applications, you have many more absorptions so you also have to consider:
Some of the more common absorbers
Xenon-135 has a very high absorption cross section, but very low availability and it is also a gas.
For shielding applications, you also have to consider common materials like water, concrete, and dirt. They don't have high absorption cross sections, but are very cheap.
Is there any isotope or particular application you had in mind?
