If a natural infection did not lead to the same response as the vaccine, the vaccine would not work. Both must result in an immune response, and antibodies. Granted, there are minor differences between them. For that matter, there are differences between patients, and between COVID variants.
No, it is not.
First you need to teach our immune system about the virus. The vaccine we used teach only one thing that we hoped should work but
The whole virus teach many things and the body has a chance to match on more information, also the information used will differ from
person to person. So Now the vaccinated will match the virus and the natural infection will match the virus and both is effective in a constant world,
yes we can think of theoretical outcomes where the only thing that is important is the same for both, but this seam not to be true as we know that
a natural infection lasts longer then the vaccine so they are _not_ the same. So to conclude the response is the same, but the dynamics
most likely different, at least the data suggest so.