Edward Jenner's vaccine wasn't a true vaccine in the modern sense. It was purposeful infection with a disease (cowpox) that, while closely related to the smallpox virus, was a milder disease than smallpox itself.
It was the last European pandemic kicked off by the Franco-Prussian war which led to the development of more modern vaccines. Actual eradication of smallpox in nature was due to a coordinated effort by the World Health Organization starting in the 1960s.