A Vaccine Question

So first, I just want to make clear: this is not some anti-vax posting. I have no general issue with vaccines, I was vaccinated as a child and teen, and I have gotten a (free) annual flu shot fairly regularly for the past decade or so.

That said, here is what is confusing me: we know (casually) a couple who is in their early 80s. They had Covid last ~ Fall. One only experienced the equivalent of a moderate cold, while the other was admitted to a hospital for 3–4 days (non-ICU). My impression was that this was an average-to-above-average experience, especially for people of their age bracket. In any case, they were fine a week or so afterwards.

Fast forward to January. They apparently both got notified that they were in the initial group eligible for vaccinations. I’m not complaining, but I’m just puzzled. If someone *had* the virus, what benefit is there for them to then get the vaccine after the fact? I have not had the flu (thankfully), but if I caught it and recovered, say, in September and the flu shot was then offered in October, I doubt I’d get it. I’m not an epidemiologist, but from all the biology classes I took, my understanding of how a vaccine works is that they introduce non-virulent / sterile forms of the virus such that the recipient’s body can then formulate antibodies against a future infection by the virus. If that’s the case, don’t you already have that protection by virtue of having had the *actual* virus? And if so, why should you get vaccinated (at least within some short window of time)?

Again, I’m not arguing in any way against vaccinations in general; I’m just trying to understand (scientifically) what the benefit is of a vaccine to a person who has recently recovered from the actual virus…?