the longbow has a longer range, and a better rate of fire. the bow was easy to disassemble, so the bowstring could be taken off in the rain, to protect it.
the crossbow was more powerful, but fired a heaver bolt, meaning there was less range. it was susceptible to failure in the rain, as the string could not be easily removed, and it took a lot more effort to prepare it for fire, by winching the bowstring onto its trigger after each shot.
However, the crossbow also could be learnt far quicker and easier than the longbow, which took years of training and constant practice to be good with - which meant that while the crossbow itself was more complex to make, its use was cheaper for training soldiers.