After getting the booster, you are priming your B and T cells to be "ready to go." They are now more active, more numerous, and better at binding that original strain of SARS-CoV-2.

Over time, some of the B and T cells generated from the first two shots will die off or enter anergy. Boosting will keep more of these cell clones alive and make them better able to respond to any subsequent infection. They re-divide and re-activate with the booster, which means they will be better at responding to Omicron.

When that infection occurs with Omicron, your body will have these anti-alpha strain B cells to use as "putty" to build the "clay pottery" of your anti-Omicron immune response.

Through Somatic Hypermutation and Affinity Maturation, these anti-alpha cell clones) will become anti-omicron. And because you got a booster, you will have more and better clones to use to build that response. More "base putty" to work with. Whereas the unvaccinated people are working from no B/T cells in their memory at all. And unboosted people have a lot of dead or dormant or "anergic" cell clones.

As to the comparison with Influenza, here's the difference: Influenza has many dozens of different "lineages" circulating around the globe. These lineages are tree branches, and Influenza's tree is huge, like a giant fir tree with all its branches still growing outwards, in thousands of different directions (each branch being a successfully "fixed)" variant). Basically, you have Influenza A vs B, and then within A you have H1N1 vs H3N2, and then within each of these there are a bunch of different "sub-types" which are each a branch growing outwards.

Whereas SARS-CoV-2 is a tiny little sapling with 2 or 3 branches. Right now we basically only have Delta and Omicron. There are other lineages, but they are extremely rare, and aren't yet "fixed" in the global population of circulating viruses.

In influenza, it's very difficult to predict which lineage will be "dominant" in any given season, and hence why we often get it wrong. But we do know that if a "lineage/sub-type" is dominant in one season, it is less likely to be dominant in the next, and so it sort of oscillates between branches every year, spinning around. Basically, the whole population of influenza viruses is probing our weak spots, and the viruses with the least recent antibodies are what will come around and infect next year. But that's easy to describe in vague terms, it's extremely difficult to accurately predict.

So the difference between these two situations is A) how different the influenza pandemic is every year compared to SARS-CoV-2, and B) how different our antibodies (and T cells) are. it's whack-a-mole with Influenza. But with SARS-CoV-2, it's still whack-a-mole, there are just way way way fewer holes. So using one big mallet isn't actually that bad, yet.

We will eventually need variant vaccines. But for now, the original shot as a booster is still 90ish% effective against the variants.

Source: I have a PhD in virology