In this case, both are doing the same except Western way is more

All viruses are species-specify, so human viruses usually only infect human, and cow viruses target cows. But in some rare cases, they can cross species boundary and infect others. It happened that  cowpox is such an exception, it can also infect human but the with reduced virulence. Since it is a close relative to smallpox, the human version of the same virus, it will trigger immune response in human and the antibodies generated by cowpox can effectively protect human against smallpox.

Once you understand how it works, it is easy to figure out what to do.