Depends on the vaccine.
For things like mumps, measles, German measles, and polio, the vaccine is probably as good as natural. Those vaccines use the whole virus. You make antibodies and T cells that recognize the entire viral coat, not just the business end. So when you “see” it again, you mount an immune response to the entire viral protein coat.
For recombinant or mRNA vaccines, your just immune to the business end. If the business end mutates too much, the immunity you have won’t work as well whereas if you have immunity to the entire virus, it can work better since the whole thing won’t mutate at the same rate.