I'm a doc and I can explain this a little more, because I just explained this to my wife the other day. I'm not vaccinated, by the way. I would probably take the Covaxin or the Novovax vaccines, but they aren't available in the US. But that's a different discussion.
The reason why these vaccines are not great against mutations is because they only protect against one specific configuration of one specific part of the virus. If you look at the anatomy of a virus, you'll see that it is much more than just spikes. There are lots of other proteins in the viral envelope.
So, let's just say that Corona virus only had 3 different parts that could be recognized by the immune system to generate a response, A1-B1-C1. Let's say that A1 is the spike protein. If you get infected with the actual virus, you now should have a natural immunity to A1, B1 and C1. If you get the vaccine only, you are only immune to A1.
Now, hit the mutate button. Now your virus is A1-B2-C1. Lucky us. Both vaccinated and non-vaccinated people (who were exposed) should be immune. Because the A1 is triggering the response in the vaccinated. And the A1 and C1 are triggering a response in the previously exposed. And, likely, there is some weak response to B2 because it is similar but not exactly matched to B1.
Know hit the mutate button again. Now your virus is A2-B2-C1. Only the previously exposed person has a strong response. Because they were exposed to C1. The vaccinated person might have a weak response to A2, but no response to B2 or C1. The problem, though, is that inside the vaccinated person, they possibly still have cells that have been hijacked by the vaccine (I don't know if they actually know how long those altered cells live). So, your body is thinking it is infected with A1 and diverting resources to fighting a virus that isn't there. So, theoretically, the vaccinated person might fare worse than the naturally exposed person.
Now hit the mutate button again. Now your virus is A2-B2-C2. The vaccinated person is in the same boat as before. And while the naturally exposed person might be there with them, the fact is that maybe one of the three mutated proteins is more similar to its original form, thereby producing a stronger response. And/or each of those things is producing a minor response. Either way, the naturally exposed person is more likely to produce a stronger response than the vaccinated only person.
But here's the fun about this. Viruses mutate to survive and evade the human immune system. If everybody is suddenly immune to A1, I can guarantee you that, in coronaviruses, the one thing that will mutate for sure is A1. It has to occur in order to survive. The vaccine will be rendered partially or completely ineffective (likely partially). And the predominant virus will be replaced by a mutant. Whether that mutant is better or worse for humans is anybody's guess. But it looks like it's probably better (from an IFR standpoint) if we can believe the PHE report.