To be sterilizing a vaccine must prevent infection. Since you never get infected you never replicate the virus and thus do not shed it. If you do not shed it the potential path of the viral life-cycle for that particular infection ends with you and thus you cannot pass on or cause a mutation. You are sterile against that disease; from the point of view of the virus you are a lifeless rock. Among commonly-used sterilizing vaccines are MMR (measles, mumps and rubella), Varicella (chicken pox), OPV (oral polio) and others. The only time that such a vaccine fails is when you do not build immunity (such as due to immune compromise.) This is extremely rare and the protection from such vaccines tends to be either decades-long or lifetime.
A vaccine that is not sterilizing permits the virus to infect you and replicate and as a result you can infect others. Technically it is not a vaccine at all (which by definition prevents infection); it is a prophylactic therapy. Such a "vaccine" instead acts to reduce or eliminate symptomatic disease. You don't know you're sick and you don't get sick. You don't go to the hospital and you don't die. Unfortunately since you don't know you're sick but are infected and the virus is both replicating in you and shedding you are more-likely to spread the infection to others. All of the current Covid jabs are in this category and so is, for that matter IPV (injected polio vaccine -- the original Salk discovery.)
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