I went back and read the description of the movie. They've got one thing totally wrong. Smallpox is not infectious until the rash comes out, by which point the person who is sick is feeling utterly lousy. This means they are not very likely to be running around spreading the disease.
Why was smallpox epidemic in the past? People living in extremely crowded and unsanitary conditions. Sharing beds and bedding with no way to do laundry or clean. Under conditions like that, disease would spread rapidly.
Did the vaccine save us from this deadly disease? Quarantine was just as important. Mass vaccination was not used to eliminate smallpox. The strategy was quarantine and ring vaccination: isolate the ill, vaccinate the contacts. Because smallpox is easily diagnosed and fairly difficult to spread (due to the natural tendency of someone who is feeling really sick to lie down and stay lying down) quarantine is a practical method of control. Much harder to prevent the spread of, say, whooping cough. Most people with whooping cough don't even know they are sick, or don't feel sick enough to quit working. And it is easily misdiagnosed.
The thing I find most puzzling about the history of smallpox and vaccination is that the smallpox vaccine isn't made with smallpox. They use a different disease substance, originally supposed to be cowpox. However, the current vaccine has been compared to cowpox and it isn't the same stuff. No one knows precisely what it is, and why it prevents a different disease. Weird.
It is also one of the most dangerous and reactive vaccines around.