Missions give a larger than normal boost to generals as well as a small boost to HP.
Crimes are essentially the same thing as Missions, only that D-rank Crimes are not available. Ranked crimes are available only as an outlaw.
It's better to do surgery than missions.
For every mission battle, there is a chance of being attacked by an earth golem instead. He deals 25% of your HP in damage and his weakness is senbon, which will hit for 5,000,000 and kill him. In B and C missions, he will give 325 exp and 0.4 generals instead. Non-Golem mission enemies will reflect metal damage (such as senbon and the enchanted sai) as a OHKO.
Generally, most users do not do D-Missions because they do not give as many generals on average as regular training.
These and higher missions will pay out more general stats than regular training. As a bonus, they'll also grant HP. Although the rewards are not as high as B-ranks, they are easier to accomplish. From this rank on, the missions are battles. Be aware there is also a 15 second timer you must wait before going back into battle while doing missions.
Instead of having to fight one enemy, you will have to fight three. The last enemy deals %-based damage.