How are vaccines and inoculations different?
If you became infected before the advent of vaccination, you would either live or die. You would also probably die with really good odds. Being said, someone who lived a disease developed immunologic tolerance to the causative agent, that much you could begin to understand. To expose someone to smallpox was to really endanger their lives, so they found that a similar virus known as Vaccinia, which would normally cause cowpox, could be administered to a patient and they would catch cowpox, but now they were resistant to catching smallpox. You can read about all this on the wiki page for vaccination.

Inoculation generally differs because in the case of inoculaton, you could hope for a less virulent strain of a pathogen but you weren't guaranteed it. You ended up with vaccination methods that increased the safety of the process: On one hand, you could infect someone with related pathogen that causes a milder disease for protection from a more dangerous, related pathogen. On the other, you could find a way to disable the agents pathogenicity, such as heat-killing. 

Modern vaccinations are much more refined in that we know so much more. Attenuated strains, for example, you take a virus that grows well in human cells and infect monkey cells just as an example. After many generations the virus grows well in the monkey cells, but no longer the human cells. The strain should still retain it's antigens, however, so when you inject a human now, the virus doesn't cause a disease but you produce antibodies to the virus.

In short, you can protect a population from a disease that would normally devastate it with a dramatically improved efficacy.