Diseases make people with them more likely to die by reducing their health (which is also the stat used to determine how likely they are to recover (does not apply to all diseases) and has a large influence on how long they'll live if they aren't killed). Additionally, most diseases lower other stats and fertility. Except for Generic Illness ("Ill"), Pneumonia, and Leprosy, all diseases can be "epidemics", which affect whole counties and can spread to characters in that county. When a county has an epidemic, the tax income is substantially reduced, I think the levies are reduced, supply limits are greatly reduced, can the outbreak can spread to neighboring counties.
Bubonic Plague kills the hell out of everyone (or at least tries to) by reducing all afflicted characters' health by 7. The most health any normal character should be born with is 6, so that means that unless the victim is Strong, their health will be reduced to zero, making them much more likely to die. So that it doesn't actually kill everyone, there is no health requirement for recovery.