There's a fairly limited number of illnesses that neutering/spaying will prevent. Obviously any diseases of the removed organs are prevented because those organs aren't there anymore. For females this is actually a pretty big deal as pyometra (infection of the uterus) is quite common in older unspayed bitches. It can be deadly if not treated, and the treatment is an emergency spay. Further, having puppies is not always the soft-focus beautiful experience that most people assume it is. A lot can go wrong and the bitch and/or all the puppies can die. If you don't breed your bitch of course this isn't going to happen, but if you have a dog who lives 14 years and you don't spay her and she comes in to heat twice a year from 6 months of age onwards, that's 27 opportunities (of a couple weeks each) for someone to make a mistake and not watch her in a situation where a male could get to her and *boom* pregnant.
In males, the only major illness that is prevented through neutering is testicular cancer. No testicals, no testicular cancer. However, un-neutered males can also get very very reckless if they are not supervised and contained at all times because if they smell a bitch in season nearby, they will do whatever it takes to get to her. Including crossing busy highways, jumping through plate glass, and jumping out of moving vehicles. These are all clearly fairly dangerous activities.
The biggest reason I see to spay and neuter is behavioral (dogs want to have sex, we do not want them to have sex, it's very frustrating for all involved) and to prevent unwanted puppies. We've got way too many as it is, we don't need to be making any more.