Coronaviruses are very different. We've rarely had any this deadly before. All vaccines are different; they're not one blanket thing. Smallpox vaccines are attenuated virus (a virus that has mutated to be benign in humans; in the case of Smallpox, the first vaccine by Jenner was injecting people with Horsepox).
Most Influenza vaccines are heat treated/inactivated virus.
In all cases, vaccines try to produce antibodies and force our immune systems to develop a memory for a virus we haven't actually been exposed to. As shown in the links above, this might not work at all for coronavirus.
To some extent, the actual interactions of vaccines, are somewhat opaque. This In A Nutshell video shows a very small part of the adaptive immune system, greatly simplified:
https://www.youtube.com/watch?v=BSypUV6QUNw
> The virus isn't just going to "burn out" anytime soon
Exponential growth doesn't go on forever. It has to hit an inflection point:
https://www.youtube.com/watch?v=Kas0tIxDvrg
The more deadly a virus is, the faster it does burn out. It runs out of hosts to infect or everyone develops an immunity. SARS/MERS may have had lower outbreaks because of how much more dangerous they were.
If you graph fatalities on a logarithmic scale, you'll see we're already well past the inflection point in the US:
https://aatishb.com/covidtrends/?data=deaths
The cases are going up, but the fatalities are not, indicating there may be a lot of over-counting. Keep in mind people who test positive but die of a car wreck are counted as COVID19. All these people getting surgery after 3 months are required to get tested just for their surgery, and a positive antibody test counts as an active case, even if they're not sick or have the virus at all.
It's so bad in the UK they've stopped reporting fatality numbers due to issues with the data.