Boron is almost exclusively an absorber, so the boron nucleus captures the neutron and it basically disappears. Hydrogen is primarily a moderator, so it reduces the energy via elastic scattering, but a significant number of very low energy neutrons still escape (some absorption to produce deuterium also occurs). Thermal neutrons can still cause damage, but they're easy to block with a subsequent thin layer of lead or something like that.
Boron's probability to capture a neutron is astronomically high, that's why you can get away with so little. Environmental sources of neutrons are actually pretty rare normally and most neutrons you do see will be pretty low energy and won't have a huge amount of penetrative power. A thin layer of boron will pretty much stop them. Pyrex (like the stuff baking dishes are made of, which is borosilicate glass - glass with boron added) is actually commonly used as control material in nuclear reactors.
Boron's probability to capture a neutron is astronomically high, that's why you can get away with so little. Environmental sources of neutrons are actually pretty rare normally and most neutrons you do see will be pretty low energy and won't have a huge amount of penetrative power. A thin layer of boron will pretty much stop them. Pyrex (like the stuff baking dishes are made of, which is borosilicate glass - glass with boron added) is actually commonly used as control material in nuclear reactors.