They are very similar. They are both workplace safety policies. They both reduce danger to other employees. They both place conditions on entering the workplace based on whether or not you put something into your body. You could say the same thing about a "no intoxication at work" policy, especially if work involves heavy machinery.

Whether it's mandating what goes into your body or what doesn't, the aim of the policy is the same: Preventing the reckless behavior of one person from harming other employees.

EDIT because I can't reply:

I don't understand what you mean by "give up control of your body". There is no evidence that taking a COVID vaccine results in loss of control of your body or limbs, or remote control of it by someone else. Hundreds of millions of people have taken it, and they all remain fully in control of their bodies.

I agree to disagree then.

The No Smoking policy mandates what you should not do, same as there is a Must Wear Clothes policy.

On the other hand you have the vaccine mandate, which forces you to give up control of your body if you want the job.

Such a policy should not be mandated by a private company, at most it should be encouraged and optional.