Why does a coffee-cup calorimeter measure $\Delta H$ while a bomb calorimeter measures $\Delta U$?

D. The coffee cup is open at constant pressure; the bomb is sealed at constant volume

At constant pressure $q_p = \Delta H$; at constant volume no expansion work is possible, so $q_v = \Delta U$. The apparatus decides which quantity you measure, which is why combustion data often needs converting between the two.